# Is the universe static?

David F. Crawford  Astronomical Society of Australia, 44 Market St., Naremburn
NSW, 2065, Australia

###### Abstract

A fundamental property of an expanding universe is that any time dependent
characteristic of distant objects must appear to scale by the factor $(1+z$).
This is called time dilation. Light curves of type Ia supernovae and the
duration of Gamma-Ray Bursts (GRB) are the only observations that can directly
measure time dilation over a wide range of redshifts. An analysis of raw
observations of 2,333 type Ia supernovae light-curves shows that their widths,
relative to a standard template, have a power-law exponent as a function of
${(1+z)}$, of (0.083 +/- 0.024) which is consistent with no time dilation and
inconsistent with standard time dilation. In addition, it is shown that the
standard method for calibrating the type Ia supernovae light curves (SALT2) is
flawed, which explains why this lack of time dilation has not been previously
observed.

Nearby observations show that the peak absolute magnitude of type Ia
supernovae is also constant. Here it is shown that the peak absolute magnitude
is independent of redshift if a static universe cosmology, Curvature
Cosmology, is used to provide the distance moduli. Furthermore it is explained
why the modified $\Lambda$-CDM model provides similar results.

Analysis of the duration of GRB shows that they are consistent with no time
dilation and have no support for standard time dilation. Consequently, this
paper argues for a fundamental change from the current paradigm of an
expanding universe to one for a static universe. Some of the major
consequences of Curvature Cosmology are listed.

###### keywords:

cosmology: large-scale structure of universe,gamma-ray bursts: general, stars:
supernovae

##  1 INTRODUCTION

Nearby type Ia supernovae are well known to have essentially identical light
curves that make excellent cosmological probes. It is argued in Section 3 that
the only characteristics of the light curve that changes with redshift are the
scaling parameters of peak luminosity and width. In particular the width which
must vary with redshift in exactly the same way as time dilation and the peak
absolute magnitude should be independent of redshift. Investigation of the
variation of the peak magnitude with redshift requires a cosmological model to
provide the distance moduli. This is done by using Curvature Cosmology
(Crawford, 2010) which will be described later (Section 7).

#### Time dilation

It is convenient to assume that the width dependence on redshift can be
described by a parameter $\alpha$ such that the redshift dependence of time
dilation is equal to $(1+z)^{\alpha}$ and then to estimate $\alpha$ from the
observations as a test of time dilation. For all current expansion models
$\alpha$ must be one and for static models, it must zero. Any significant
difference of $\alpha$ from either of these two values could only be explained
by a completely new cosmology.

#### Photon energy

In Section 2 it is argued that in quantum mechanics the apparent wavelength of
a photon is a measurement of its energy and as a consequence its redshift may
be due to any process that causes a loss of energy. Thus in quantum mechanics,
the rigid nexus between the shift of spectral lines and other time variations
is broken.

#### Intrinsic widths

The observational evidence for standard time dilation has a long history with
notable papers being Goldhaber et al. (2001) and Blondin et al. (2008). The
observed width of any light curve from a distant object is the product of an
intrinsic width, with a cosmological width due to time dilation. If the
observed wavelength is $\lambda$ then the intrinsic wavelength is
$\epsilon=\lambda/(1+z)$ which is shorter than the observed wavelength. Since
many of the intrinsic wavelengths are beyond the visible range, their
intrinsic widths cannot be easily determined from nearby supernovae. A
suitable method of solving this problem is to generate a reference template
from all the supernovae light curves that provides a complete light curve for
each intrinsic wavelength, and then to use these templates to accurately
calibrate the observations by eliminating any intrinsic effects.

#### Salt2

The SALT2 method (Guy et al., 2007, 2010) determines these intrinsic templates
by combining a large number of observations over a wide range of redshifts and
has been used by Betoule et al. (2014), Conley et al. (2011), Foley et al.
(2018), Scolnic et al. (2017) and Jones et al. (2018). In other words, the
reference template is the average of the light curves from many supernovae as
a function of intrinsic wavelength. Consequently the measurement of the
intrinsic width at a particular intrinsic wavelength can come from many
supernovae with different redshifts.

It has been first shown by Crawford (2017) and here in Section 4 that there is
a fundamental problem with the SALT2 analysis in that a systematic variation
in width as a function of redshift is included in the template as a systematic
variation of the width as a function of intrinsic wavelength. The SALT2
calibration process is very good at removing the intrinsic variations but at
the same time, it removes systematic redshift variations such as time
dilation. Thus supernovae light curves that have been calibrated by SALT2, or
a similar method, have all the cosmological information that is a power-law
function of redshift removed.

#### Type Ia supernova light curve widths

A major part of this paper (Section 5) is an examination of the raw
observations for 2,333 supernovae to investigate how the widths of their light
curves vary with redshift. This was done separately for each filter so that
observations of each supernova can provide up to five distinct and unrelated
values for the width of the observed light curve.

The first step of the analysis is to determine the intrinsic width as a
function of intrinsic wavelength and the second step is to use the intrinsic
width to remove the intrinsic component from the raw observations in order to
obtain a cosmological width that is entirely due to cosmological processes.

#### Type Ia supernovae absolute magnitudes

It has also been observed that nearby type Ia supernovae have peak absolute
magnitudes that are independent of redshift. The aim here is to extend this
analysis of peak absolute magnitudes to higher redshifts. However there is a
major problem in that the estimation of the absolute magnitude requires a
distance modulus to convert the apparent flux density to an absolute magnitude
and this requires the use of a cosmological model. This analysis is done for
two different models: the standard $\Lambda$-CDM model (summarised in Appendix
C) and a static model, Curvature Cosmology (Section 7).

The first step of the analysis is to get the average absolute intrinsic
magnitude as a function of intrinsic wavelength. Then to remove these
intrinsic effects from the observations to provide cosmological peak absolute
magnitudes for each filter and each supernova. This is done, separately, for
both cosmological models.

#### Gamma- Ray Bursts

Gamma-Ray Bursts (GRB) are the only other observations that can provide direct
measurements of time dilation over a wide range of redshifts. The analysis
used here in Section 6 investigates the observed durations of the bursts as a
function of redshift and again finds that they are consistent with no time
dilation.

#### Curvature Cosmology

Following the analysis a brief description of Curvature Cosmology is provided
(Section 7) as well as the description of the cosmological consequences of
Curvature Cosmology (Section 7.10).

#### Basic assumptions

It is assumed for this analysis that the intrinsic properties of the type Ia
supernovae light curves and the GRB burst durations are the same at all
redshifts. In other words, we are assuming that there is no evolution. Part of
this assumption for type Ia supernovae is that minor differences in the
subtypes and effects of the host galaxy do not have a significant dependence
on redshift. Hence their main effect is to increase the background noise.
Furthermore it is assumed that the wavelength dependence of a filter can be
replaced by a single value at the effective wavelength of the filter.

##  2 REDSHIFTS AND TIME DILATION

The Hubble redshift law states that distant objects appear, on average, to
have an apparent velocity of recession that is proportional to their distance.
Since this is consistent with models in General Relativity that have universal
expansion, such expanding models have become the standard cosmological
paradigm. Classically, this redshift was obvious because in these models
spectral lines are shifted in wavelength exactly like any other time-dependent
phenomena.

However quantum mechanics tells us that light is transmitted by photons whose
effective wavelength is determined from their momentum by the de Broglie
equation $\lambda=hc/E$ where $E$ is their energy and $\lambda$ is their
effective wavelength. Thus their effective wavelength is simply a measurement
of their energy and is not a proper wavelength in the classical sense.
Nevertheless it does describe how photons can be diffracted and their energy
measured by an interferometer. The Doppler effect and the universal expansion
are explained by an actual loss (or gain) of photon energy. A consequence is
that redshifts may be due to any process that causes a loss of photon energy.
Thus in quantum mechanics, the rigid nexus between the shifts in wavelength of
spectral lines and other time variations is broken.

##  3 COSMOLOGICAL CHARACTERISTICS OF LIGHT CURVES

Let us assume that the intrinsic radiation characteristics of type Ia
supernovae are independent of redshift and that in an expanding universe the
rate of universal expansion is constant for the duration of the light curve.
Since cosmology only controls the transmission of the light, it follows that
the shape of the received light-curve must be the same as the shape of the
intrinsic light curve but with different scale factors. In other words, the
cosmology can only change the peak flux density and the width of the light
curve. Consequently, all of the cosmological information is contained in the
dependence of these two variables with redshift. Thus, it is only necessary to
measure these two scaling parameters in order to investigate the cosmology of
supernovae light curves. Furthermore, we only need determine these two
parameters for intrinsic light curves in order to distinguish them from
cosmological effects.

Since the observed light curve is the intrinsic light curve multiplied by any
time dilation, then the observed width is the product of the intrinsic width
and the time dilation width. We assume that the time dilation width has the
power law of ${(1+z)}^{\alpha}$, where $\alpha$ is the exponent and has a
value of one for standard time dilation. By definition, the redshift, $z$, is
defined by $\epsilon=\lambda/{(1+z)}$, where $\lambda$ is the observed
wavelength and $\epsilon$ is the intrinsic (emitted) wavelength.

The observed wavelength is determined by the filter used and the redshift of
the supernova is usually measured from the observed wavelength shift of
emission or absorption lines. The intrinsic width is a function of the
intrinsic wavelength and we assume for part of this work that it has the power
law $\epsilon^{\beta}$, where $\beta$ is determined by observations. Although
the intrinsic light curve width almost certainly has a more complicated
function of wavelength, it is only that part of it that can be described by
this power law that will enable it to be confused with time dilation. Hence
the model used here is that the observed width, $w(\lambda)$ is

$w(\lambda)={(1+z)}^{\alpha}\epsilon^{\beta},$ (1)

where the reference width is one. Substituting for $\epsilon$ provides the
more informative equation

$w(\lambda)=\lambda^{\beta}{(1+z)}^{(\alpha-\beta)}.$ (2)

This shows the close relationship between intrinsic and cosmological widths.

Note that there is a common intrinsic variation in either light curve width or
absolute peak magnitude for all filters. This intrinsic curve replaces all the
k-corrections, colour corrections, and similar methods used to correct
observations to standard filters or redshifts.

##  4 INTRINSIC DEPENDENCE OF LIGHT CURVES

Observations of local type Ia supernovae show that the emission from the
expanding gas cloud is multicoloured and the intensity is a function of both
wavelength and time. A major practical problem is that the emitted wavelengths
are often much shorter than the observed wavelengths and since the shape and
size of the intrinsic light curve is a function of the wavelength the analysis
of observations requires that this intrinsic dependence is known. For high
redshift supernovae, many of the emitted wavelengths are outside the visual
range, which means that we cannot, in general, use nearby supernovae to obtain
the required calibrations.

An ingenious solution, exemplified by the SALT2 method (Guy et al., 2007,
2010), is to determine the calibration spectra from averaging the light curves
of many supernovae at many different redshifts. Because the only observations
available are from filters that cover a large wavelength range this is a
difficult process. This and similar methods carefully deconstruct the average
light curve, as a function of intrinsic wavelengths from a large number of
observations, and then generate a light-curve template for each intrinsic
wavelength. Thus the light curve for any particular intrinsic wavelength will
have contributions from supernovae at many observed wavelengths.

###  4.1 A flaw in the SALT2 method

However, there is a problem first described by Crawford (2017) with the SALT2
method of determining the characteristics of the intrinsic light curve. Let
$w(\lambda)$ be the observed width at, $\lambda$, and let $W(\epsilon)$ be the
width at the intrinsic (rest frame) wavelength $\epsilon$. (The use of $w$ and
$W$ was chosen to mimic the familiar use of $m$ and $M$ for magnitudes.)
Similarly, let $f(\lambda,z)$ and $F(\epsilon,z)$ be the observed and emitted
flux densities.

Equation 2 shows that there is a close correspondence between a systematic
variation in width with intrinsic wavelength and time dilation. Although this
is interesting, the extension to a wide range of wavelengths needs a more
refined analysis.

The supernovae observations typically consist of the flux-density measure in
filters that essentially cover the visual wavelengths. Since each filter has a
filter gain function, $g(\lambda)$ which is the fraction of power transmitted
per unit wavelength, then the flux density observed by a particular filter at
the wavelength $\lambda$ is given by

$f(\lambda,z)\propto\int g(\lambda^{*})F(\epsilon,z)\,d\lambda^{*},$ (3)

where $\lambda^{*}$ is a dummy integration variable. Now the width of the
light curve can be determined by measuring the difference between the two half
peak flux density points. To the first order, this process is a linear
function of the flux densities and will certainly be sufficiently linear in
the last step of an iterative process of measuring the width. Thus we can
apply Eq. 3 to the widths to get

$w(\lambda,z)=\int g(\lambda^{*})W(\epsilon)\,d\lambda^{*}.$ (4)

Now suppose the intrinsic light curves have a power-law wavelength dependence
so that $W(\epsilon)=\epsilon^{\beta}$ where $\beta$ is a constant. Then
including this power law in Eq. 4 gives

$w(\lambda,z)=\int
g(\lambda^{*})\left(\frac{\lambda^{*}}{1+z}\right)^{\beta}\,d\lambda^{*}.$ (5)

Since the $(1+z)$ term is independent of $\lambda^{*}$ it can be taken outside
of the integral to get

$w(\lambda,z)=A(1+z)^{-\beta},$ (6)

where

$A=\int g(\lambda)\lambda^{\beta}\,d\lambda.$ (7)

Clearly, $A$ is only a function of $\beta$ and the filter characteristics and
it is the same for all observations with this filter. Consequently, if the
intrinsic widths have a power-law dependence on wavelength, proportional to
$\epsilon^{\beta}$ and from Eq. 6, this will be seen as a power-law dependence
of the observed widths that is proportional to $(1+z)^{-\beta}$.

Conversely, if there is no intrinsic variation of the widths of the light
curve with wavelength but there is a time dilation with exponent $\alpha$,
then the derived intrinsic wavelength dependence in the SALT2 template from
multiple supernovae will have a power-law dependence with $\beta=-\alpha$.

In practice, this means that during the generation of a reference spectrum,
any observed time dilation is recorded in the templates as an intrinsic
wavelength dependence. When this is used to calibrate new observations that
(by definition) have the same time dilation, this redshift dependence in the
observations will be cancelled by the wavelength dependence in the template
and the calibrated widths will be independent of redshift.

Consequently, if the SALT2 or similar calibration method is used then any
cosmological information that was in the calibration observations in the form
of a power law of $(1+z)$, will be removed from subsequent analyses. Simply
put, the SALT2 calibration removes all power laws as function of $(1+z)$,
whether artificial or genuine, leaving the calibrated light curve without any
of this power-law information and therefore without any cosmological
information.

###  4.2 SALT2 template analysis

In order to remove the expected time dilation, the first step of the standard
SALT2 analysis is to divide all the epoch differences by ${(1+z)}$. If there
is no time dilation, this will produce an effective time dilation of
${(1+z)}^{-1}$. Figure 1 shows the relative width (in black and yellow points)
for each intrinsic wavelength of the light curves in the SALT2 template (cf.
Appendix A). Since there are clearly problems with some of the widths, shown
in yellow, the analysis was confined to the black points. The explanation for
the bad widths is unknown but one contributing factor could be poor data for
wavelengths between the filters. Shown in Figure 1 are some filter response
curves for the nearby supernovae where this effect would be most pronounced.

Figure 1:  A plot, in black, of the relative widths of the light curves for
the SALT2 templates as a function of intrinsic wavelength. The blue line is
the best fit power law with an exponent $1.240\pm 0.014$. Yellow points are
assumed to be invalid. For comparison some of the filter response curves for
nearby supernovae are also shown.

The blue line is the best power-law fit of the black points and has an
exponent of $\beta=1.240\pm 0.011$. Then allowing for the initial division of
the epoch differences by ${(1+z)}$ either there is no time dilation
($\alpha=0$) and an intrinsic dependence with exponent $\beta=0.240\pm 0.011$,
or that it has the standard time dilation ($\alpha=1$) and a large intrinsic
dependence with exponent $\beta=1.240\pm 0.011$. Since the initial division of
the epoch differences by ${(1+z)}$ could produce the slope $\beta=1.0$ this
analysis shows strong support for zero time dilation.

Note that if there is no time dilation and the effects of auxiliary parameters
are small, then the SALT2 stretch factors are estimates of the true width.

##  5 THE ANALYSIS OF TYPE Ia SUPERNOVAE LIGHT CURVES

###  5.1 The raw observations

Crawford (2017) describes the selection and analysis of the original
observations of type Ia supernovae light curves that have been selected by
Betoule et al. (2014), who have provided an update of the Conley et al. (2011)
analysis with better optical calibrations and more supernovae. This JLA (Joint
Light-curve Analysis) list sample has supernovae from the Supernova Legacy
Survey (SNLS),nearby supernovae (LowZ), the Sloan Digital Sky Survey (SDSS)
(Holtzman et al., 2008; Kessler et al., 2009) and those observed by the (HST)
(Hubble Space Telescope) (Riess et al., 2007; Jones et al., 2013). Also
included are 1169 supernova from the Pan-STARRS supernova survey (Kaiser et
al., 2010; Jones et al., 2018; Scolnic et al., 2018). The sources of the raw
observations are listed in the Appendix A.

For each type Ia supernova, the data used here was, for each filter, a set of
epochs with calibrated flux densities and uncertainties. The observations
taken with the $U$ and $u$ filters are very noisy, and following Conley et al.
(2011) and Betoule et al. (2014), the observations for these filters were not
used. Table 1 shows the number of type Ia supernovae, number of accepted
supernovae, and number of accepted filter data for each survey. Most of the
missing filter sets were because they did not have at least one observation
prior to five days before the peak flux density epoch and at least one five
days after the peak.

Table 1: Surveys and numbers Survey | Nsupernovae | Naccepted | Nfilters
---|---|---|---
LOWZ | 241 | 96 | 290
SDSS | 625 | 485 | 1563
SNLS | 252 | 196 | 463
PS1MD | 1169 | 731 | 1362
HST | 46 | 6 | 6
Total | 2333 | 1514 | 3684

###  5.2 Type Ia supernovae reference template

The essential aim of this analysis is to determine $\alpha$ and $\beta$ in Eq.
1, by examining the raw observations of type Ia supernovae. A critical part of
any investigation into type Ia supernovae light curves is to have a reference
template. In order to remove any possible bias, a standard independent
template, the $B$ band Parab-18 from Table 2 Goldhaber et al. (2001) which has
a half-peak width of 22.3 days has been used. Then the procedure is (for each
supernova) to determine the observed width of the light curve for each filter,
relative to the template light curve, and then estimate $\alpha$ and $\beta$
from all the widths as a function of redshift.

###  5.3 The analysis of raw observations

As an example of a typical type Ia supernova Figure 2 shows the light curves
for four filters for the SNLS supernova SN2007af with the filters used being
shown in the legend (Goobar & Leibundgut, 2011). Accepted data points are
shown as coloured symbols whereas the rejected points are shown with an open
square.

The first feature to notice is that the epoch of the peak flux density depends
on the filter type and is therefore a function of the intrinsic wavelength.
Secondly, there is a secondary peak at about 25 days after the first peak for
the longer wavelength filters. Although this second peak is intrinsic to the
supernova, it does not appear to be very consistent (Elias et al., 1981;
Meikle & Hernandez, 2000; Goobar & Leibundgut, 2011). Consequently, as shown
in Figure 2, all filters, except $B$, $j$, and HST, had their epochs more than
15 days after the main peak rejected.

Figure 2: The light curves for the SNLS type Ia supernova SN2007af. Valid
points are shown as full squares and invalid points as open squares. To avoid
confusion, the filter results have been vertically displaced, The secondary
peak is clearly apparent.

Unfortunately, the direct analysis of the data to obtain the epoch of the peak
flux density, the value of the peak flux density and the light curve width
using a $\chi^{2}$ method has an intrinsic problem in that position of the
peak flux density and the width are not completely independent. However the
value of the peak flux density is almost independent of the width estimate.

The first step was to estimate the value of the peak flux density using a
minimum $\chi^{2}$ procedure. Next the program uses the reference light curve
and the ratio of the flux density to the peak flux density to obtain a flux
density epoch. This epoch has an uncertainty equal to the flux density
uncertainty divided by the absolute value of the slope of the reference light
curve at that epoch. Then a simple weighted regression of the observed epochs
verses the flux density epochs provides the peak flux density offset and the
width of the light curve. This estimate of the peak flux density offset was
ignored. However it was found by minimising the standard $\chi^{2}$ for the
flux densities that was calculated using the estimates for the widths and the
flux densities.

This method has the bonus of providing uncertainty estimates for the widths.
Note that each supernova had separate values for the peak flux density and the
width for each filter.

One problem that was noticed is that the range of the uncertainties in the
flux densities for many filters was too large. There were 336 cases out of
42,818 filter sets where the ratio of the smallest uncertainty to the largest
uncertainty was less than 10%. The problem is that one of these very precise
flux densities could have a weight one hundred times larger than other flux
densities which could produce anomalous results. The observation method for
most of these supernovae is to observe the same patch of sky with the same
telescope and settings for each epoch. Although there can be nights with bad
seeing, the expected uncertainty in the flux density is the same for each
observation. Consequently all the flux density uncertainties were replaced by
a common value of 3% of the peak flux density.

After all the parameters were estimated for a particular filter and the
supernova, the flux density for each epoch was tested to see if it was an
outlier. This was done by computing a value

$l_{i}=f_{i}/f_{peak}-h_{i},$ (8)

where for each epoch, $f_{i}$ is its flux density, $f_{peak}$ is the peak flux
density, $h_{i}$ is the height of the reference light curve at that epoch, and
$i$ is its index. Then an epoch it was rejected as an outlier if the value
$|l_{i}-\bar{l}|$, where $\bar{l}$ is the average for all the other epochs
$l_{i}$, was greater than five times the rms for all the other epochs. The
epoch with the largest discrepancy was eliminated, then a full analysis was
repeated and this continued until there were no more outliers. Out of 30,850
accepted epochs, there were 964 (3%) outliers.

The major selection criteria for each valid supernova was that, for each
filter, there was at least one epoch less than five days from the peak epoch
and one that was five days greater than the peak epoch, and that there were at
least 4 valid epochs. And in order to show a reasonable fit to the light curve
the width uncertainty must be greater than 0.005 and less than 0.3.

###  5.4 Light curve widths

Figure 3: The $\chi^{2}$ fit for light curve widths as a function of $\alpha$.
For each $\alpha$ the intrinsic width is set to be $\beta=\alpha-0.352$.

fig4fig5

Figure 4:  This is a plot of the observed widths of type Ia the light curves
as a function of intrinsic wavelength and redshift. The symbol and colour for
each filter are shown in the legend. The black line is the average value as a
function of $\epsilon$. For the right figure, the black line shows the width
for $\alpha=0$ and the red line shows the width for standard time dilation
$\alpha=1$. The symbol and colour for each filter are shown in the legend. In
order to avoid clutter only the HST observations (shown in black) have
uncertainty estimates.

Because of Eq. 2 there is confusion in measuring $\alpha$ and $\beta$ from the
raw observations. However Eq. 2 shows that a regression of the logarithm of
the width verses the logarithm of $(1+z)$ will provide an estimate of the
combined $\alpha-\beta$. The result for this raw width is

$\alpha-\beta=0.324\pm 0.025.$ (9)

One way to determine which time dilation is appropriate for these observations
is to plot the $\chi^{2}$ of the fit as a function of $\alpha$ where for each
$\alpha$ we put $\beta=0.324-\alpha$, thus preserving the raw width. The
$\chi^{2}$ function is

$\chi^{2}=\sum_{i}[\log(w_{i})-A-\alpha\log(1+z_{i})-\beta\log(\lambda/(1+z_{i}))]^{2},$
(10)

where the summation is over all the observations with index $i$, $w_{i}$ is
the width and $A$ is a dummy variable that is determined for each $\alpha$.
The result is shown in Figure 3. Clearly the best fit is when $\alpha\approx
0.2$ which is compatible with zero time dilation which implies that
$\beta\approx-0.124$.

This is supported by another estimate of $\beta$ from the average value of the
logarithm of the ratio of the widths of two filters from the same supernovae.
Although this method assumes that both filters have the same width it is
independent of the value of the width. For 496 supernovae that had valid
observations in the $g$ and $z$ filters the estimate is $\beta=-0.302\pm
0.023$.

A more general and better analysis is to assume that $\alpha=0$ and the get
the intrinsic widths a function of $\epsilon$ rather than restrict them to a
simple function $(1+z)^{\beta}$. This is easily done by averaging the widths
for each value of $\epsilon$. Figure 4 shows a scatter distribution of
intrinsic width for each observation together with its average value. For
reference purposes the value of the average exponent $\beta$ was also
estimated. For 3,684 accepted widths the result was $\beta=0.212\pm 0.007$.
This value is in excellent agreement with the SALT2 value of $\beta=0.240\pm
0.011$(Section 4.2).

The final step is the estimate $\alpha$ by a regression of the logarithm of
the raw widths corrected for intrinsic width verses $log(1+z)$. The correction
was done by dividing the observed width by the intrinsic width. The regression
equation is

$w_{cosmological}=(-0.044\pm 0.004)(1+z)^{0.083\pm 0.024}.$ (11)

This width is self-consistent with zero which strongly favours a static
universe with no time dilation. Figure 4 shows the intrinsic distribution and
the plot of the cosmological widths for 3,684 filters from 2,333 type Ia
supernovae.

###  5.5 Redshift dependence of the light curve widths

The observed light curve for each of four redshift ranges was computed for
each epoch in the range from -15 days to 40 days from the peak flux-density
epoch. This was done by selecting all relative flux densities that were within
one day of this epoch and setting the value of the light curve to be the
median of these selected flux densities. The median was used because it is
insensitive to extreme values. Although this method is using the same data,
its advantage is that it depends only on the relative flux density for each
epoch and does not depend on the fitting procedure. This is similar to the
type of analysis done by Goldhaber et al. (2001) and Blondin et al. (2008).

Figure 5:  The type Ia supernovae light curves for four redshift ranges for
the static model. The legend shows the colour for each redshift range. The
template light curve is shown in black. The epochs have been corrected for the
intrinsic width by multiplying each epoch difference by the appropriate
intrinsic width

The results are presented graphically in Figure 5 which shows the average
light curve for four ranges of redshift. The black curve shows the master
template light curve. Table 2 shows the redshift range, the mean redshift, the
number of points, and the average width for each range. Note that the observed
width for each epoch has been corrected for its intrinsic component by
multiplying the epoch distance from the peak flux density epoch by the
intrinsic width show by the black line in Figure 4.

Table 2: Light curve widths for four redshift ranges Range | $\bar{z}$ | number | Width
---|---|---|---
0.00-0.15 | 0.069 | 25 | $0.903\pm 0.005$
0.15-0.30 | 0.225 | 25 | $0.954\pm 0.009$
0.30-0.50 | 0.383 | 25 | $1.059\pm 0.024$
0.50-1.30 | 0.649 | 25 | $1.195\pm 0.033$

The power law fit for these four widths with respect to $(1+z)$ has an
exponent of $0.023\pm 0.012$ which has a negligible dependence on redshift.

###  5.6 Type Ia supernovae peak magnitudes

fig7fig8

Figure 6:  Scatter plots of the intrinsic absolute magnitude as a function of
$\epsilon$ and the cosmological peak absolute magnitude as a function of
redshift for the $\Lambda$-CDM model. The black line in the left figure shows
the average value of the intrinsic magnitude.

fig9fig10

Figure 7:  Scatter plots of the intrinsic absolute magnitude as a function of
$\epsilon$ and the cosmological peak absolute magnitude as a function of
redshift for Curvature Cosmology. The black line in the left figure shows the
average value of the intrinsic magnitude.

Since the observed light curve is the intrinsic light curve multiplied by time
dilation, then the observed flux density is the product of the intrinsic flux
density and the cosmological scaling factor. There is strong evidence from
nearby supernovae that the expected peak absolute magnitude is the same for
each supernova. Clearly, this commonality requires a valid distance modulus
coming from a valid cosmological model. The analysis is done for the standard
$\Lambda$-CDM cosmology and for a static cosmology. A suitable static model is
Curvature Cosmology which is a complete cosmology that shows excellent
agreement with all major cosmological observations. A brief description of the
cosmology is given later (Section 7).

For convenience all the flux densities were converted into AB magnitudes. The
analysis procedure used is identical to that used above for the light curve
widths which starts with the measurement of the intrinsic absolute magnitudes
which are then subtracted from the raw magnitudes to get the cosmological
magnitudes. Table 3 shows the results for both cosmologies for the exponent
$\alpha$ from the regression of the absolute peak magnitudes verses
$-2.5\log_{10}(1+z)$. The top row is for the raw peak magnitudes and the
bottom row is for the cosmological peak magnitudes.

Table 3: Peak magnitude exponents Parameter |  $\Lambda$-CDM | Curvature Cosmology
---|---|---
$\alpha_{raw}$ | $0.182\pm 0.030$ | $-0.385\pm 0.043$
$\alpha_{cosmological}$ | $-0.014\pm 0.030$ | $0.050\pm 0.030$

Both models are consistent with the peak absolute magnitude being independent
of redshift. Since the type Ia supernovae observations have been a major
contribution the $\Lambda$-CDM model it is not surprising that it has a good
fit to this data.

The good fit of Curvature Cosmology is a strong endorsement of this model.
Note that it was formulated long before good supernovae observations became
available and its distance modulus has no fitted parameters except for $H$
(Eq. 20) which an additive constant. Figure 6 shows the intrinsic absolute
magnitude as a function of $\epsilon$ and the cosmological peak absolute
magnitude as a function of redshift for the $\Lambda$-CDM model. Figure 7
shows the same results for Curvature Cosmology.

##  6 GAMMA RAY BURSTS

The website of the Neil Gehrels Swift Observatory, which runs the Swift
satellite, that contains the Burst Alert Telescope (BAT) describes them as:
“Gamma-ray bursts (GRBs) are the most powerful explosions the Universe has
seen since the Big Bang. They occur approximately once per day and are brief,
but intense, flashes of gamma radiation. They come from all different
directions of the sky and last from a few milliseconds to a few hundred
seconds.” An important characteristic of the BAT is that it has a photon
counting detector (Barthelmy et al., 2005) that detects photons in the 15-150
keV energy range with a resolution of about 7 keV. It can also image up to 350
keV without position information. An important parameter for each burst is T90
which is a measure of the burst duration. The start and end times of T90 are
defined as the times the fraction of photons in the accumulated light-curve
reaches 5% and 95%.

The Third Swift Burst Alert Telescope Gamma-Ray Burst Catalog (Lien et al.,
2016) states that “Many studies have shown that the observed burst durations
do not present a clear-cut effect of time dilation for GRBs at higher
redshift.” Indeed the upper panel of their Figure 25 shows that there is no
obvious trend of the burst length with redshift except for a decrease in the
number of short bursts with larger redshifts. This shows some support for the
“tip-of-the-iceberg” effect which is sometimes used to explain the lack of
strong time dilation in the GRB durations. However, there is no obvious change
in the duration of longer bursts with redshift.

Now the number distribution of photons in GRB bursts is close to a power law
with an exponent of about -1.6. As the redshift increases, the number of
detectable photons will rapidly decrease as many more photons will be below
the detector limit. If we assume that the distribution of photons as a
function of energy is independent of the position of the photon in the burst,
then there should be no expected change in T90 with redshift. On the other
hand, if the higher energy photons are clustered towards the centre of the
GRB, then the intrinsic T90 should decrease with increasing redshift.
Consequently, we would expect to see the normal time dilation or maybe a
little less in the T90 measurements.

This analysis directly examines the exponent of a power-law regression of
measured T90 of raw GRB data (c.f. Appendix B), that had burst durations above
2 seconds, as a function of $(1+z)$(Lien et al., 2016). Since there were no
T90 uncertainties provided, the analysis used an unweighted regression. The
power-law fits were done for the T90 duration with the exponent shown in row 1
of Table 4 which is consistent with no time dilation. The problem with this
and similar analyses is that the variables have a very large scatter in values
which would require very large numbers of GRB to achieve absolutely conclusive
results.

In a recent analysis Zhang et al. (2013) claim that the GRB T90 widths are
consistent with an expanding universe. They measured T90 in the observed
energy range between $140/(1+z)\,$keV and $350/(1+z)\,$keV, corresponding to
an intrinsic energy range of $140-350\,$keV. Their exponent for these selected
is $0.94\pm 0.26$ which is consistent with the standard expanding model.

My reanalysis of their raw, T90 widths using the data in their Table 1, is
shown in columns 2 and 3 of Table 4 and are consistent with no time dilation.
The Swift and Zhang et al. (2013) unselected T90 widths are displayed in
Figure 8. Although they have many common GRB, there are small differences in
the T90 widths. This is because Zhang et al. (2013) have used their own
analysis of the original data to get their own values for T90 widths.
Examination of Figure 8 shows the large scatter of the T90 widths and it also
shows that they are consistent with no time dilation and are unlikely to be
consistent with standard time dilation.

My determination of the exponents of their energy selected widths as a
function of $(1+z)$ is shown in rows 4 and 5 of Table 4. The unweighted result
in row 4 agrees with their result. However the exponent for the weighted
analysis shown in row 5 is consistent with no time dilation.

Table 4: Exponents for redshift dependence of GRB

Row | Data | Weighta | N | Exponent
---|---|---|---|---
1 | Swift | U | 298 | $0.39\pm 0.17$
2 | Zhangb | U | 139 | $0.10\pm 0.26$
3 | Zhangb | W | 139 | $-0.16\pm 0.20$
4 | Zhangc | U | 139 | $0.94\pm 0.26$
5 | Zhangc | W | 139 | $0.31\pm 0.23$

a U denotes an unweighted fit, W denotes weighted

b Raw T90 from Zhang et al. (2013) in their Table 1

c T90,z for intrinsic energy range of 140-350 keV.

The use of energy selection for T90 implies that there is an intrinsic
dependence of burst duration on the photon energy. Since the BAT has a photon
counting detector, any measurement of T90 is independent of the selected
photon energies. The only restriction is that the photon energies must be
within the detector limits. Thus BAT does not have the energy selection that
is necessary for this analysis. Furthermore, it is difficult to understand how
any subset of photons that are detected can have a different time dilation
from the rest of the photons in the same GRB. If we ignore the energy-selected
Zhang et al. (2013) results, the conclusion is that the burst length of GRB is
consistent with no time dilation and has very little support for the standard
model.

Figure 8: A plot of T90 as a function of redshift. The green line shows the
line for no time dilation, the red line shows the line for standard time
dilation, and the black line shows a time dilation with the fitted exponent of
$0.39$. The Swift data are shown in blue filled circles and the Zhang (Zhang
et al., 2013) data are shown as red diamonds.

##  7 CURVATURE COSMOLOGY

Curvature Cosmology (Crawford, 1987a, b, 1991, 1993, 1995a, 1995b, 1999, 2006,
2009b, 2009a, 2010) is a complete cosmology for a static universe that shows
excellent agreement with all major cosmological observations without needing
dark matter or dark energy.(Note that (Crawford, 2010) is an update with
corrections of the previous work.) This cosmology depends on the hypotheses of
Curvature Redshift and Curvature Pressure described below.

The basic cosmological model is one in which the cosmic plasma dominates the
mass distribution and hence the curvature of space-time. In this first-order
model, the effects of galaxies and stars are neglected. The geometry of this
cosmology is that of a three-dimensional surface of a four-dimensional hyper-
sphere. It is almost identical to that for Einstein’s static universe. For a
static universe, there is no ambiguity in the definition of distances and
times. One can use a cosmic time and define distances in light travel times or
any other convenient measure. Curvature Cosmology obeys the perfect
cosmological principle of being statistically the same at all places and at
all times.

###  7.1 Curvature Redshift

The derivation of Curvature Redshift is based on the fundamental hypothesis of
Einstein’s general theory of relativity that space-time is curved. As a
consequence, for positive curvature, the trajectories of initially parallel
point particles, geodesics, will move closer to each other as time increases.
Consequently the cross-sectional area of a bundle of geodesics will slowly
decrease.

In applying this idea to photons, we assume that a photon is described in
quantum mechanics as a localised wave where the geodesics correspond to the
rays of the wave. Note that this wave is quite separate from an
electromagnetic wave that corresponds to the effects of many photons. It is
fundamental to the hypothesis that we can consider the motion in space-time of
individual photons. Because the curvature of space-time causes the focussing
of a bundle of geodesics, this focussing also applies to a wave. As the photon
progresses, the cross-sectional area of the wave associated with it will
decrease. However, in quantum mechanics properties such as angular momentum
are computed by an integration of a radial coordinate over the volume of the
wave. If the cross-sectional area of the wave decreases, then the angular
momentum will also decrease. However, angular momentum is a quantised
parameter that has a fixed value.

The solution to this dilemma is that the photon splits into two very low-
energy photons and a third that has the same direction as the original photon
and nearly all the energy. It is convenient to consider the interaction as a
primary photon losing a small amount of energy to two secondary photons. This
energy loss will be perceived as a small decrease in frequency. By symmetry
the two secondary photons with identical energies are emitted at right angles
to the trajectory, which means that there is no apparent angular scattering.

Since in quantum mechanics electrons and other particles are considered as
waves, a similar process will also apply. It is argued that electrons will
interact with curved space-time to lose energy by the emission of very low-
energy photons.

From Crawford (2010) we get the basic equation for the fractional change in
energy of the photon. This is based on the equation of geodesic deviation
(Misner et al., 1973).

$\frac{1}{E}\frac{dE}{ds}=-\left(\frac{8\pi
G\rho}{c^{2}}\right)^{1/2}=-1.366\times 10^{-13}\sqrt{\rho}\,\mbox{m}^{-1}.$
(12)

For many astrophysical types of plasma, it is useful to measure density by the
equivalent number of hydrogen atoms per cubic metre: that is we can put
$\rho=$N MH and get

$\frac{1}{E}\frac{dE}{ds}=-\sqrt{\left(\frac{8{\pi}GNM_{H}}{c^{2}}\right)}=-5.59\times
10^{-27}\sqrt{N}\,\mbox{m}^{-1}.$ (13)

The rate of energy loss per distance travelled depends only on the square root
of the density of the material, which may consist of gas, plasma, or dust. For
many astrophysical plasmas the frequency of the emitted photons will be less
than the plasma frequency and they will be absorbed and heat the plasma.

Another important factor is that if there is any other competing interaction
which occurs before the secondary photons are produced it will inhibit the
Curvature Redshift. Such an interaction is the coherent multiple scattering
that produces refractive index. This can be important for ground bases
experiments and for radio frequency observations in the Galaxy. For example,
most lower frequency radio observations in our galaxy will be unaffected by
Curvature Redshift.

###  7.2 Curvature Pressure

The hypothesis of Curvature Pressure is that for moving particles there is a
pressure generated that acts back on the matter that causes the curved space-
time. In this case, Curvature Pressure acts on the matter that is producing
curved space-time in such a way as to try to decrease the curvature. In other
words, the plasma produces curved space-time through its density entering the
stress-energy tensor in Einstein’s field equations and the actions of the
velocities of the plasma particles is to try and decrease this curvature..

A simple cosmological model using Newtonian physics illustrates some of the
basic physics subsequently used to derive the features of Curvature Pressure.
The model assumes that the universe is composed of gas confined to the three-
dimensional surface of a four-dimensional hyper-sphere. Since the
visualisation of four dimensions is difficult let us suppress one of the
normal dimensions and consider the gas to occupy the two-dimensional surface
of a normal sphere. From Gauss’s law (i.e. the gravitational effect of a
spherical distribution of particles with radial symmetry is identical to that
of a point mass equal in value to the total mass situated at the centre of
symmetry) the gravitational acceleration at the radius $r$ of the surface is
normal to the surface, directed inward and it has the magnitude
$\ddot{r}=-GM/r^{2}$ where $M$ is the total mass of the particles and the dots
denote a time derivative. For equilibrium, and assuming all the particles have
the same mass and velocity we can equate the radial acceleration to the
gravitational acceleration and get the simple equation from celestial
mechanics of

$v^{2}=\frac{{GM}}{{r}}.$

The effect of this balancing of the accelerations against the gravitational
potential is seen within the shell as a Curvature Pressure that is a direct
consequence of the geometric constraint of confining the particles to a shell.
If the radius $r$ decreases then there is an increase in this Curvature
Pressure that attempts to increase the surface area by increasing the radius.
For a small change in radius in a quasi-equilibrium process where the particle
velocities do not change the work done by this Curvature Pressure (two
dimensions) with an incremental increase of area $dA$ is $p_{\rm c}dA$ and
this must equal the gravitational force times the change in distance to give

$p_{\rm c}dA=\frac{{GM^{2}}}{{r^{2}}}\,dr,$

where $M=\sum{m_{i}}$ with the sum going over all the particles. Therefore,
using equation (7.2) we can rewrite the previous equation in terms of the
velocities as

$p_{\rm c}dA=\frac{{M\left\langle{v^{2}}\right\rangle}}{r}\,dr.$

Now $dA/dr=2A/r$, hence the two-dimensional Curvature Pressure is

$p_{\rm c}=\frac{{M\left\langle{v^{2}}\right\rangle}}{{2A}}.$

Thus in this two-dimensional model the Curvature Pressure is like the average
kinetic energy per unit area. This simple Newtonian model provides a guide as
to what the Curvature Pressure would be in the full General Relativistic
model.

The extension to different particle masses and velocities uses the basic
property of General Relativity that gravitation is an acceleration and not a
force. This is supported by Eötvös, Pekár & Fekete (1922), Dicke (1964), and
Braginskij & Panov (1971) who have shown that the passive gravitational mass
is equal to the inertial mass to about one part in $10^{12}$. The usual
interpretation of this agreement is that they are fundamentally the same
thing. However, an alternative viewpoint is that the basic equation is wrong
and that the passive gravitational mass and the inertial mass should not
appear in Newton’s gravitational equation. Consequently Newton’s gravitational
equation is an equation of accelerations and not of forces. The equation for
Curvature Pressure in a 3 dimensional high temperature plasma is

$p_{\rm c}=\frac{1}{3}\left\langle{\gamma^{2}-1}\right\rangle\rho c^{2},$ (14)

where $\gamma$ is the Lorentz factor and $\langle\rangle$ denotes an average.

In effect, my hypothesis is that the cosmological model must include this
Curvature Pressure as well as thermodynamic pressure. Note that although this
has a similar form to thermodynamic pressure it is quite different. In
particular, it is proportional to an average over the squared velocities and
the thermodynamic pressure is proportional to an average over the kinetic
energies. This means that, for plasma with free electrons and approximate
thermodynamic equilibrium, the electrons will dominate the average due to
their much larger velocities.

Including Curvature Pressure into the Friedmann equations provides stable
static cosmological model. Including Curvature Pressure from Eq. 14 the
modified Friedmann equations are

$\displaystyle\ddot{R}$ $\displaystyle=$ $\displaystyle-\frac{4\pi
G\rho}{3}\left[{1-\left\langle{\gamma^{2}-1}\right\rangle}\right]R,$
$\displaystyle\dot{R}^{2}$ $\displaystyle=$ $\displaystyle\frac{8\pi
G\rho}{3}R^{2}-c^{2}.$

Clearly, there is a static solution if $<\gamma^{2}-1>=1$, in which case
$\ddot{R}=0$. The second equation, with $\dot{R}=0$ provides the radius of the
universe which is given by

$R=\sqrt{\frac{3c^{2}}{8\pi G\rho}}{\mbox{\, }}=\sqrt{\frac{3c^{2}}{8\pi
GM_{\rm H}N}}.$ (15)

Thus, the model is a static cosmology with positive curvature. Although the
geometry is similar to the original Einstein static model, this cosmology
differs in that it is stable. The basic instability of the static Einstein
model is well known (Tolman, 1934; Ellis, 1984). On the other hand, the
stability of Curvature Cosmology is shown by considering a perturbation
$\Delta R$, about the equilibrium position. Then the perturbation equation is

$\Delta\ddot{R}\propto\left(\frac{d\langle\gamma^{2}-1\rangle}{dR}\right)\Delta
R.$ (16)

For any realistic equation of state for the cosmic plasma, the average
velocity will decrease as $R$ increases. Thus the right-hand side is negative,
showing that the result of a small perturbation is for the universe return to
its equilibrium position. Thus, Curvature Cosmology is intrinsically stable.
Of theoretical interest is that Eq. 16 predicts that oscillations could occur
about the equilibrium position.

###  7.3 X-ray background radiation

Since Giacconi et al. (1962) observed the X-ray background, there have been
many suggestions made to explain its characteristics. Although much of the
unresolved X-ray emission comes from active galaxies, there is a part of the
spectrum between about 10 keV and 1 MeV that is not adequately explained by
emission from discrete sources.

Curvature Cosmology can explain the X-ray emission in the energy range from
about 10 keV to 300 keV as coming from a very hot intergalactic plasma. A
simple model has a mixture of hydrogen with 8% helium and a measured density
of $N=1.55\pm 0.01$ hydrogen atoms per cubic metre or 2.57$\times
10^{-27}\,$kg m-3.

For this density the predicted a temperature is $2.56\times 10^{9}\,$K for the
cosmic plasma. The temperature estimated from fitting the X-ray data is
$(2.62\pm 0.04)\times 10^{9}\,$K which is a good fit. Although this is similar
to early explanations of the X-ray emission, it differs in that it depends on
the current plasma density. The earlier explanations required the X-ray
emission to come from a plasma with about three times that density which
conflicted with other observations.

###  7.4 Nuclear abundances

One of the successes of the standard model is in its explanation of the
primordial abundances of the light elements. In Curvature Cosmology, the
primordial abundance refers to the abundance in the cosmic plasma from which
the galaxies are formed.

The first point to note is that the predicted temperature of the cosmic plasma
is $2.56\times 10^{9}{\mbox{ K}}$ at which temperature nuclear reactions can
proceed. It is postulated that there is a continuous recycling of material
from the cosmic gas to galaxies and stars and then back to the gas. Because of
the high temperature, nuclear reactions will take place whereby the more
complex nuclei are broken down to hydrogen, deuterium, and helium.

###  7.5 Cosmic microwave background radiation

The Cosmic microwave background radiation (CMBR) is produced by very high
energy electrons via Curvature Redshift radiation in the cosmic plasma. With
$N=1.55$ the predicted temperature of the CMBR is 3.18 K to be compared with
an observed value of 2.725 K (Mather et al., 1990). This prediction does
depend on the nuclei mix in the cosmic plasma and could vary from this value
by several tenths of a degree.

Although the CMBR photons are subject to continuous Curvature Redshift they
will be quantised, and since all energy levels are freely available, the black
body (Plank function) is their thermal equilibrium spectrum.

Differences in the local environment, especially high density lower
temperature gas clouds, will decrease the flux density of the CMBR and could
explain some of the observed spatial fluctuations in the CMBR.

###  7.6 No Dark matter and galactic rotation

In 1937 Zwicky (1937) found in an analysis of the Coma cluster of galaxies
that the ratio of total mass obtained by using the virial theorem to the total
luminosity was 500 whereas the expected ratio was 3. The virial theorem is a
statistical theorem that states that for an inverse square law the average
kinetic energy of a bound system is equal to half the potential energy. This
huge discrepancy was the start of the concept of dark matter. It is surprising
that in more than eight decades since that time there is no direct evidence
for dark matter. Similarly the concept of dark energy (some prefer
quintessence) has been introduced to explain discrepancies in the observations
of type 1a supernovae.

X-ray observations show that the Coma cluster has a large plasma cloud in its
centre. The Curvature Cosmology model is that the galactic velocity dispersion
in the cluster is entirely due to Curvature Redshift of photons passing
through the central plasma cloud. For 583 galaxies the rms (root-mean-square)
velocity was 893 km s-1 and the computed theoretical value was $554$ km s-1.
Considering that it was assumed that both the galaxy distribution and plasma
distribution had a very simple geometries this shows that Curvature Cosmology
can explain the velocity dispersion in the Coma cluster.

One of the most puzzling questions in astronomy is: why the observed velocity
of rotation in spiral galaxies does not go to zero towards the edge of the
galaxy. Simple Keplerian mechanics suggest that there should be a rapid rise
to a maximum and then a decrease in velocity that is inversely proportional to
the square root of the radius once nearly all the mass has been passed.
Although the details vary between galaxies, the observations typically show a
rapid rise and then an essentially constant tangential velocity as a function
of radius out to distances where the velocity cannot be measured due to lack
of material. The standard explanation is that this is due to the gravitational
attraction of a halo of dark matter that extends well beyond the galaxy.

Observations show that our own Galaxy and other spiral galaxies have a gas
halo that is larger than the main concentration of stars. It is clear that if
the observed redshifts are due to Curvature Redshift acting within this halo,
the halo must be asymmetric; otherwise, it could not produce the asymmetric
rotation curve. Now the observed velocities in the flat part of the curves are
typically 100 to 200 km s-1. For realistic values of the densities and sizes
of the halo, the velocity is about 163 km s-1. Thus, the magnitude is
feasible. Although there could be a natural asymmetry in a particular galaxy,
the fact that the flattened rotation curve is seen for most spiral galaxies
suggests that there is a common cause for the asymmetry. One possibility is
that the asymmetry could arise from ram pressure due to the galaxy moving
through the intergalactic medium.

Although the explanation for galactic rotation observations is limited, Coma
cluster observations show no support for dark matter. Since Curvature
Cosmology can explain all the supernova observations there is nor support for
dark energy.

###  7.7 No Black Holes

A theory of Curvature Pressure in a very dense medium where quantum mechanics
dominate and where general relativity may be required is needed to develop
this model. Nevertheless it is clear that Curvature Pressure would resist a
hot compact object from collapsing to a black hole. Because of the potential
energy released during collapse, it is extremely unlikely for a cold object to
stay cold long enough to overcome the Curvature Pressure and collapse to a
black hole.

What is expected is that the final stage of gravitational collapse is a very
dense object, larger than a black hole but smaller than a neutron star. This
compact object would have most of the characteristics of black holes. Such
objects could have large masses and be surrounded by accretion discs. Thus,
many of the observations that are thought to show the presence of a black hole
could equally show the presence of these compact objects.

If the compact object is rotating there is the tantalising idea that Curvature
Pressure may produce the emission of material in two jets along the spin axis.
This could be the ”jet engine” that produces the astrophysical jets seen in
stellar-like objects and in many huge radio sources. Furthermore this could be
a mechanism to return material to the cosmic plasma. Currently there are no
accepted models for the origin of these jets.

###  7.8 Olber’s Paradox

In Curvature Cosmology Olber’s Paradox is not a problem. Visible light from
distant galaxies is shifted into the infrared where it is no longer seen and
the energy is eventually absorbed back into the cosmic plasma. Everything is
recycled. The plasma radiates energy into the microwave background radiation
and into X-rays. The galaxies develop from the cosmic plasma, stars are formed
which pass through their normal evolution. Eventually all their material is
returned to the cosmic plasma.

###  7.9 Basic equations for Curvature Cosmology

The geometry is that of a three-dimensional surface of a four-dimensional
hyper sphere. For this geometry the radius is $r=R\chi$ where

$\chi=\ln(1+z)/\sqrt{3}.$ (17)

(NB. work prior to 2009 has $\chi=\ln(1+z)/\sqrt{2}$))
The area is

$A(r)=4\pi R^{2}\sin^{2}(\chi).$ (18)

The surface is finite and $\chi$ can vary from 0 to $\pi$. The volume within a
redshift $z$ is given by

$V(z)=2\pi R^{3}\left[{\chi-\frac{1}{2}\sin(2\chi)}\right].\\\ $

Using the density $N=1.55\,{m}^{-3})$ the Hubble constant is predicted to be

$\displaystyle H$ $\displaystyle=$
$\displaystyle-\frac{c}{E}\frac{{dE}}{{ds}}=\left(8\pi GM_{\rm
H}N\right)^{\frac{1}{2}}$ $\displaystyle=$ $\displaystyle
51.69N^{\frac{1}{2}}\;\mbox{kms}^{-1}\;\mbox{Mpc}^{-1}$ $\displaystyle=$
$\displaystyle 64.4\pm 0.2\;\mbox{kms}^{-1}\;\mbox{Mpc}^{-1}.$

The only other result required here is the equation for the distance modulus
($\mu=m-M$), which is

$\mu=5\log_{10}[(\sqrt{3}\sin(\chi))/h]+2.5\log_{10}(1+z)+42.384.\\\ $ (20)

where h=H/(100 $\mbox{kms}^{-1}\;\mbox{Mpc}^{-1}$).

###  7.10 Basic consequences of Curvature Cosmology

Since the ramifications of a static universe are quite profound, a list of the
major consequences of Curvature Cosmology is given here. All the numerical
results are derived using the cosmic plasma density $N=1.55$ H atoms m-3.

  1. 1.

It obeys the perfect cosmological principle

  2. 2.

It is stable (Section 7.2).

  3. 3.

There is no dark matter. (Section 7.6)

  4. 4.

There is no dark energy. Meaningless.

  5. 5.

There is no inflation. Meaningless.

  6. 6.

There is no horizon problem. Meaningless.

  7. 7.

The cosmic plasma has a density N$=1.55\pm 0.01\,$M${}_{H}\,{m}^{-3})$.

  8. 8.

The cosmic plasma has a temperature of $(2.64\pm 0.04)\times 10^{9}$ K.

  9. 9.

The Hubble constant is $64.4\pm 0.2\;\mbox{kms}^{-1}\;\mbox{(}Mpc)^{-1}$.

  10. 10.

It is consistent with supernovae observations.

  11. 11.

It is consistent with GRB observations.

  12. 12.

It is consistent with quasar luminosity observations.

  13. 13.

It is consistent with galaxy luminosity observations.

  14. 14.

It is consistent with Tolman surface brightness observations.

  15. 15.

It is consistent with radio source counts.

  16. 16.

It is consistent with quasar variability

  17. 17.

It is consistent with angular size observations.

  18. 18.

It is can explain the Cosmic Microwave Background Radiation (Section 7.5).

  19. 19.

The CMBR radiation has temperature of $3.18$ K (Section 7.5.

  20. 20.

It is can provide a partial explanation for fluctuations in CMBR.

  21. 21.

It is can provide a partial explanation for galactic rotation curves (Section
7.6).

  22. 22.

It is can explain the X-ray background radiation (Section 7.3).

  23. 23.

It is can possibly explain the cosmic nuclear abundances (Section 7.4).

  24. 24.

Curvature redshift can be investigated with laboratory measurements.

  25. 25.

There are no black holes (Section 7.7).

  26. 26.

Universal radius: $3.11\times 10^{26}$ m or $1.008\times 10^{10}$ pc.

  27. 27.

Volume: $8.95\times 10^{80}$ m3 or $2.02\times 10^{31}$ pc3.

  28. 28.

Mass: $2.54\times 10^{54}$ kg or $1.28\times 10^{23}$ $\cal M_{\bigodot}$.

Of interest is that in Curvature Cosmology (CC) distant objects will always
have a fainter absolute magnitude than for the standard model. Table 5 shows
the distance moduli for both cosmologies, their difference, and the absolute
flux density ratio for a range redshifts.

Table 5: Relative absolute magnitudes Redshift |  $\Lambda$-CDM | CC | diff. | ratio
---|---|---|---|---
0 | 42.394 | 42.394 | 0.000 | 1.000
1.0 | 43.512 | 43.067 | 0.445 | 1.507
2.0 | 45.189 | 44.418 | 0.771 | 2.035
5.0 | 47.425 | 45.078 | 1.447 | 3.792
10.0 | 49.102 | 46.927 | 2.175 | 7.412
20.0 | 50.750 | 47.629 | 3.122 | 17.729
50.0 | 52.884 | 48.050 | 4.834 | 85.859
100.0 | 54.468 | 47.682 | 6.786 | 517.962

##  8 CONCLUSION

The first part of this paper argued that the only effect of cosmology on
supernovae light curves is to change the scaling parameters of peak flux
density and width. The shape of the light curve is intrinsic to the supernovae
and is unchanged by cosmology.

Next, it was argued that the redshift of photons is a measure of their energy
and could be caused by any systematic energy loss or by time dilation.

In Section 4 and 5 it has been shown that there is a major problem in using
SALT2, and similar calibration methods, to remove the intrinsic wavelength
dependence of widths from type Ia supernovae light-curve observations. The
process of generating the templates means that if the observed light curves
have widths that contain the effects of time dilation, these effects are
incorporated into the template. The subsequent use of the template will remove
this time dilation affects whether artificial or genuine, from the new
observations.

Consequently, SALT2 calibrated light curves cannot contain any cosmological
data that is in the form of a power law. Consequently previous analyses of
type Ia supernovae gave self-consistent results because of a flaw in the
standard analysis program SALT2.

The light curve widths of type Ia supernovae are consistent with no time
dilation with an exponent of $0.083\pm 0.024$ which is completely inconsistent
with standard time dilation which means that the universe is static.

The absolute magnitudes are consistent with Curvature Cosmology with an
exponent, $\alpha$, of $-0.050\pm 0.030$, whereas the $\Lambda$-CDM has an
exponent of $-0.014\pm 0.030$. From the excellent agreement of the
$\Lambda$-CDM model it is apparent that the $\Lambda$-CDM distance modulus has
been modified to achieve this goal. This has occurred because of the strong
belief that the standard time dilation and the $\Lambda$-CDM model are both
valid.

One way to partially validate these conclusions would be to redo the SALT2
analysis without the initial division of the epoch differences by $(1+z)$.

In addition the duration of Gamma Ray Bursts are completely consistent with a
static universe.

###### Acknowledgements.

This research has made use of the NASA/IPAC Extragalactic Database (NED) that
is operated by the Jet Propulsion Laboratory, California Institute of
Technology, under contract with the National Aeronautics and Space
Administration. The calculations have used Ubuntu Linux and the graphics have
used the DISLIN plotting library provided by the Max-Plank-Institute in
Lindau.

##  Appendix A SOURCE OF SUPERNOVAE OBSERVATIONS

All of the original type Ia supernovae observations were retrieved from the
SNANA (Kessler et al., 2009) in the download package $snana.tar.gz$ on the
website http://www.snana.uchicago.edu using the index files shown in Table 6.

Table 6: Index source files for SNANA data file
---
lcmerge/LOWZ_JRK07
lcmerge/JLA2014_CSP.LIST
lcmerge/JLA2014_CfAIII_KEPLERCAM.LIST
lcmerge/SNLS3year_JRK07.LIST
lcmerge/SDSS_allCandidates+BOSS_HEAD.FITS
lcmerge/JLA2014_SNLS.LIST
lcmerge/JLA2024_HST.LIST
lcmerge/SDSS_HOLTZ08

A current SALT2 template file for the JLA (Joint Light-curve Analysis)
analysis was taken from the SNANA website in the directory
$models/SALT2/\\-SALT2/JLA\\-B14$.

The Pan-STARSS supernovae were accessed from the site
https://archive.stsci.edu/prepds/ps1cosmo/jones and the file datatable.html.

Basic information for all the filters used is shown in Table 7 where column 1
is the filter name, column 2 is the mean wavelength in $\mu\,$m, column 3 (N)
is the final number of supernovae with a valid light curve for this filter,
and column 4 is the HST filter name.

Table 7: Filter characteristics Name | Wavelength/$\mu\,m$ | N | HST
---|---|---|---
B | 0.436 | 202 |
V | 0.541 | 205 |
R | 0.619 | 130 |
I | 0.750 | 239 |
g | 0.472 | 1,933 |
r | 0.619 | 2,035 |
i | 0.750 | 2,071 |
z | 0.888 | 1,936 |
6 | 0.907 | 5 | F850LP
7 | 1.249 | 1 | F125W

##  Appendix B SOURCE OF GRB OBSERVATIONS

The raw GRB data was taken from
$https://swift.gsfc.nasa.gov/archive/grb_{t}able$ that had burst durations
longer than 2 seconds and valid measurements for the redshift, T90, the
fluence and the peak one-second photon flux rate. The data labelled “Zhang”
comes from Table 1 in Zhang et al. (2013).

##  Appendix C Equations for $\Lambda$-CDM COSMOLOGY

The equations needed for the modified $\Lambda$-CDM model (Hogg, 1999; Goliath
et al., 2001; Barboza & Alcaniz, 2008), with $\Omega_{M}=0.27$, $\Omega_{K}=0$
and where $h$ is the reduced Hubble constant, are listed below. The symbol
$w^{*}$ is used for the acceleration parameter in order to avoid confusion
with the width $w$. These equations depend on the function $E(z)$ defined here
by

$E(z)=\int_{0}^{z}\frac{dz}{\sqrt{\Omega_{M}(1+z)^{3}+(1-\Omega_{M})(1+z)^{(1+w^{*})}}}.$
(21)

The distance modulus is

$\mu_{B}(z)=5\log_{10}(E(z)(1+z)/h)+42.384.$ (22)

The co-moving volume is

$v_{B}(z)=\frac{4\pi}{3}(2.998E(z)/h)^{3}{Gpc}^{3}.$ (23)

The equation of state parameter $w^{*}$ in the expansion model distance
modulus is included to investigate the effects of including the cosmological
constant. Conley et al. (2011) found that the parameter, $w^{*}$, has a value
$w^{*}=-0.91$, whereas Sullivan et al. (2011) found that $w^{*}=-1.069$.
Although its actual value is not critical for this paper the value of $w^{*}$
is chosen to be $w^{*}=-1.11$, so that $E_{B}$ would be the best fiducial
constant with the values for the peak magnitudes and stretch factors provided
by B14.

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