It has recently been reported [1] on the apparent null effect of cosmic time dilation upon light curve power spectra measurements of some 800 low and high redshift quasars (QSO) monitored for 28 years. This appears to contradict two Supernovae Type Ia (SNe-Ia) light curve evolution studies [2] [3] which show the effect of

broadening of the power spectra time series consistent with a cosmic time dilation of for redshift z. In

this paper, we show that both the QSO and SNe-Ia results are compatible if account is made for the relativity of cosmic time as developed in the theory of Cosmological Special Relativity (CSR) [4] . We also apply these concepts to fitting the combination of high redshift SNe-Ia distance data [5] and Gamma-Ray Bursts (GRB) data [6] .

The Cosmological Relativity of Carmeli [7] , the general and special theories, is a five dimensional brane world model based on time t, space and velocity v. It makes only one new assumption, that of the maximum value of cosmic time which is the Hubble-Carmeli time constant. In the same way that the constant speed of light c constrains the observations in space and time, so too the constant of cosmic time constrains the observations of space and velocity. The familiar Lorentz transformations of Einstein Special Relativity (SR) between observers moving at constant relative velocity v carry over into Cosmological Special Relativity (CSR) between observers separated by relative cosmic time t. And just as in SR, there is the special way that velocities add together which reduces to the Galilean form at low velocities with respect to the speed of light, in CSR cosmic times add in an analogous way which has the form for low values with respect to cosmic time but is modified for larger cosmic times.

In this paper, we derive only the minimum CSR framework we require for the development of cosmic time transformation effects and the luminosity distance relation. We show that for weak fields and (to be defined below), the universe has a flat, Euclidean geometry. And, due to the additive properties of cosmic time in CSR, this gives us a unique form for the luminosity distance relation. We model cosmic time aging effects in light curve data from SNe-Ia. In comparison to the standard Friedmann-Lemaître-Robertson-Walker (FLRW) model, the CGR luminosity distance relation performs quite well in fits to SNe-Ia and GRB distance data, although it requires more dark matter in the mass density. As a consequence, larger rest masses are predicted for the hypothetical particle [8] decay products and. We also derive a relation for the first acoustic peak of the Cosmic Microwave Background (CMB) anisotropy.

The five dimensional Cosmological General Relativity of Carmeli [7] is approximated by the linearized metric element,

where

The coordinates are time t, spatials x, y and z and velocity v, with c the speed of light in vacuum and the Hubble-Carmeli time constant. The linearized model is accurate to first order in and. The parameter is the Hubble constant at zero distance and no gravity and where is the

Hubble constant. The fields and are general functions of the coordinates. The CGR standard value for h is (to three digits) [9] ,

This gives

In Cosmological Relativity the time coordinate is measured backwards from the present at to the big bang at. Just as the speed of light c is the maximum observable speed, the time is the maximum observable cosmic time. The gravitational fields are specified by the functions and. The expansion of the universe occurs when

and it is observed at a specific cosmic time t, so that

Applying (5) and (6) to (1) yields for the expanding universe

which reduces to

The metric (1) defines the Einstein field equations in five dimensions [7] (Sect. 7.3)

where is the mixed Ricci tensor [7] (Appendix A), R is the Ricci scalar and is the mixed energy-momentum tensor, where is the effective mass density and is the velocity vector. The indices for the five dimensions of. The Kronecker delta for, and for. The Carmeli gravitation constant where G is Newton's gravitation constant. The component of (9) gives us the equation

where

where and. From (10) and (12) we get the Poisson equation for cosmology, in the space-velocity domain,

The effective mass density is defined by

where is the mass density and is the critical mass density. Under the assumption that the mass is uniformly distributed, the mass density is independent of the spatial coordinate r, but can depend on time t and velocity v. The critical mass density is a constant defined by

It is useful to express the effective mass density as a parameter in terms of the critical mass density. Dividing by we have

where is the mass density parameter. For a spatially uniform mass distribution, is independent of r, the solution to (13) then takes the form ( [7] , Sect.7.3.2),

where M is an optional point mass centered at the origin of r. Assuming no central mass, we set. Then putting the expression for from (17) into (8) and simplifying we get

The integration of (18) over r and v in the space-velocity domain is carried out at a specific time t. We assume that the mass density is a function of cosmic time only, so that will be constant throughout the integration. Substitute by (16) and put (18) into the integral form

Integrating (19) and solving for r in terms of v we obtain the general solutions

By use of the identities and, where x is real and, we write the general solution as

Equation (17) gave the solution for the field. Now consider the solution for the field in (1). The component of (9) gives us the equation [7]

where

where and. Similar to the case for obtaining, we solve (26) to obtain

where M is an optional point mass centered at the origin of r. We see by (17) and (27) that in fact,

and furthermore, that the constants c and are part of the (cosmological) first term while the constants c and G are part of the (Newtonian) second term. Assuming no central mass (), for the case where the mass density becomes equal to the critical density, , from (16), the effective mass density parameter and the universe becomes Euclidean.

On the other hand, more generally, we can derive a flatspace metric. Given (28), if we divide (1) by, we can write

where

and

with the condition that

This implies that for weak fields and we can express the cosmology of the expanding universe in terms of a flat space Euclidean geometry, with a metric of the form. In the next section we derive this flat space special theory for cosmology.

By (29), with notation, we have the metric

where r is given by (31). The expansion of the universe occurs when and observations are made at a particular instant of time t so that. Then for the expansion, (33) gives us

which, upon integration gives the Hubble law

where.

In the observable universe there are two classes of objects, those that are bounded by the gravitation of their combined masses and those that are observed to be moving away from one another in the Hubble flow. In other words, if we lump all nearby neighboring galaxies into a super galaxy mass point, then the universe would consist of only super galaxy mass points flying apart in the Hubble flow. Cosmological Special Relativity (CSR) describes these super galactic objects in the universe. However, unlike SR which can have real observers in reference frames which move relatively at less than light speed, in CSR, all galaxies are in the Hubble flow and expand at the Hubble rate h. That is, there are no objects not in the Hubble flow from which to set up a frame of reference and compare observations. Consequently, for CSR we define hypothetical observers in their frames which move relatively at a rate. We derive the transformation of coordinates between these hypothetical frames. However, the coordinates of real galaxies are included in the transformation.

In CSR the age of the universe is, the Hubble-Carmeli time constant, which is assumed to be the same for all cosmic time relative inertial observers and is the maximum cosmic time (just as in SR the speed of light c is the maximum velocity and is constant for all velocity relative inertial observers.) Assume that there are hypothetical observers in reference frames K and separated by a fixed cosmic time. Each frame is assumed to be unaccelerated (inertial) with respect to cosmic time. Transformations are made between K

describing an object O with “4-vector” coordinates and describing the same object O with 4- vector coordinates. The magnitude of each 4-vector is defined by

with the invariant condition

where and. For the case that object O is a galaxy in the expansion, the invariants. We refer to S and as 4-vectors even though there are only 2 components, since the 3 spatial components are condensed into.

To obtain the cosmological transformation ( [7] , Sect. 2.2), analogous with the Lorentz transformation, assume that a linear transformation exists between the coordinates of hypothetical frames K and. In frame K define the space-velocity 4-vector with magnitude

Similarly, in frame define the space-velocity 4-vector with magnitude

By the requirement that the magnitude of a 4-vector is invariant under transformations between reference frames then

Clearly, by (39) and (40), the 4-vectors S and describe coordinates which can be either in the Hubble flow, when, or not in the Hubble flow, when. In order to obtain the transformation equations between the 4-vectors it is required that the reference frames K and be separated by a fixed cosmic time of.

Then, for constants and, where is a constant hyperbolic angle, define and such that

and

To solve for the angle use the boundary condition which represents the origin of frame. At (42) yields,

where

is the fixed cosmic time separating the origins of frame relative to K. From (44), for

where is a limiting condition, we use the hyperbolic functional identities to obtain,

Substituting from (47) and (48) into (42) and (43) we obtain

By inverting (49) and (50) we get the inverse transform equations

It can be verified that the transformations (49)-(52) satisfy the invariance requirement of (41) that by direct substitution into (36) and (37). As was previously mentioned, for a galaxy O with 4-vector observed by the observer in frame K, where, the transformation Equations (49) and (50)

gives for the galaxy O observed in the 4-vector where, by the invariance of the transformation,. Thus, Hubble coordinates are conserved. In the next section we will use this property to obtain the cosmological redshift relation.

We wish to quantify observations of light wave phenomena in the expanding universe made by observers at different cosmic times. Consider the distance r to a galaxy O which is in the Hubble flow, measured by the observer at the origin of K. For the observer at the origin of the distance to the same galaxy O is. If and, where the l’s are the measured wavelengths of the light from the galaxy and N is the fixed number of wavelengths, then taking the ratio of distances we get

where z is the cosmological redshift of the light due to the expansion of space during the cosmic time t between frames and K. Substituting for r from (51) into (53) gives

For a galaxy which is in the Hubble flow,

Substituting from (56) into (55), and along with (53) yields

Equation (57) is the cosmological redshift of the wavelength of light measured between observers in frames K and. Inverting (57) we get

This transformation is similar to the lengthening of the wavelength of light from a distant galaxy by the factor. From the cosmological redshift relation (57) and the fact that the periods and of a wave of light are related to the wavelengths and, respectively, by

we have from (53),

where and. Then the cosmic time dilation from (61) is given by

where we assume that is an arbitrary time interval in.

Dividing (51) by (52) we obtain the transformation for the addition of cosmic time from to K, analogous to the addition of velocities in SR,

where. The inverse transformation, from K to is obtained by dividing (49) by (50) giving

where. We refer to (63) and (64) as the general cosmic time addition relations.

Setting in (63) yields

implying that two cosmic times can never add up to more than. The identical result is obtained from (64) for the time when.

Combine the two transformations by taking the product of time dilation (62) due to the expansion of the universe with time contraction (69) due to the addition of cosmic times. The total elapsed cosmic time observed by the observer in K at cosmic time 0 for a small time change in frame at cosmic time t is given by

where by substituting for from (57), the cosmic aging function is defined by

Substituting for from (58) in terms of redshift z into (73) yields,

The cosmic aging function for, which gives a time dilation, and for, which corresponds to a time contraction. The maximum occurs at where which, from (57), corresponds to a redshift. Figure 1 is a plot of the cosmic aging function.

We use the distance relation (23) with the density, giving

where is the average mass density parameter at the present epoch.

In CGR the luminosity distance relation includes the contraction of a small interval of cosmic time in the past (69). It enters through the concept that the energy measured over an interval of time from a source with luminosity L at rest in frame at cosmic time t, radiates a proportional quantity of energy E measured by the observer in K given by

which from (73) is the cosmic aging function operating on the time interval. Using (76) as the form for the energy of the source in the derivation [10] , the luminosity distance is given by

which is a factor of the standard form, implying that in CGR sources appear less luminous and thus further away due to the relativity of cosmic time. Substituting for distance r from (75) and for from (57), and simplifying, (77) becomes

In practice we make the substitution, [4] (Appendix B. 4.2).

We apply the luminosity distance relation (78) plus a calculated best fit fixed offset to get the apparent magnitude of a distant luminous source,

where we apply (58) to convert from to z in obtaining. We use the CGR standard value [9] of. In practice the source absolute magnitude is absorbed into the value of the offset. We include in our analysis data from both SNE-Ia and GRB studies. The SNE-Ia data come from the Supernova Cosmology Project SCP Union 2.1 data set [5] of 580 SNe-Ia magnitudes and errors up to. The GRB distance data of 69 burst events come from [6] which are selected events up to from website data provided by [11] . For the observed magnitudes and respective errors, the for the fit is defined by

The reduced chi-squared is given by

where N is the number of data samples and k is the number of fitting parameters.

For the model the luminosity distance relation is given by

where for flat space [12] .

Although SNE-Ia data are independent of any particular cosmology, this is not so for GRB data which must be calibrated with a specified cosmological model. This is because SNE-Ia have nearby sources to use for calibration, but for GRB there are no nearby sources for this purpose. The original GRB data set was calibrated with the cosmology. To recalibrate the GRB data for the CGR cosmological model [13] we use the relation

where is the original GRB distance modulus calibrated with the model, is the distance modulus calibrated for the CGR model and is a conversion factor. The magnitude errors were also converted using (83). We determined a good fitting value of which was used to convert the GRB distance moduli for all fits to the CGR model. The converted GRB data were combined with the SNE-Ia

data to form the complete data set. For the CGR model, the number of parameters for, and. The parameter is fixed in this analysis, so the number of degrees of freedom, the same as for the model. The best fit for the CGR model is shown in Table 1 for (a conservative estimate of error) with offset and having a reduced chi-squared for. The results are shown in Figure 2. Figure

3 shows the histogram of the normalized residual errors for the fit. The solid curve is a Gaussian with mean and standard deviation with an amplitude estimated “by eye” to give a close fit to the histogram. The fit appears good [6] [13] [14] .

We fit the cosmology to the original GRB data set, which was calibrated with and. We assume parameters in the model, for, and. We use a fixed Hubble

constant of. The best fit occurred for offset. Figure 4 shows the Hubble Diagram for the fit of to the combined data, with the GRB moduli used unaltered from the

data set. The parameter is fixed in this analysis, so the number of degrees of freedom is the same as for the CGR model. The reduced chi-squared. Figure 5 shows the histogram of the normalized residual errors for the fit. The solid curve is a Gaussian with mean and standard deviation with an amplitude taken from the CGR histogram.

Under the reduced chi-squared statistical model, the ideal reduced with the errors distributed normally (Gaussian) about with. The model with a reduced which is closest to 1 is preferred. Models with reduced are deemed to have too few parameters, so “under fit” the data. Models with reduced are deemed to have too many parameters and thus “over fit” the data.

We see that the CGR model under fits the data by, while the model over fits the data by. For this analysis, the CGR model is times better at fitting the combined data. However, considering that we did not vary the mass densities and, nor the Hubble parameter, when fitting with the model, we make this conclusion as mainly a statement of our confidence in the CGR model. A more rigorous analysis of the fitting operation would be required.

Assume that the mass density, that is, composed of baryonic matter and cold dark matter. For (95% confidence level) [15] , with the CGR value, this gives

Then, for and, the CGR model gives a dark matter density of

compared to the model which gives a dark matter density of

In an extension to the standard model (SM), a hypothetical X particle [8] is theorized to exist, having 2 species, and (and their conjugate species and.) These particles were generated non-thermally during the early universe. The decays either into a visible three quark state (UDD), or the hidden state (), with each state having baryon number +1. The conjugate state decays to the visible three quark state () or the hidden state (), with each state having baryon number −1. All of the dark matter today, in

this extended model, is theorized to be composed entirely of hidden particle states (). Rare processes can transfer baryonic number from the hidden sector to the visible sector through inelastic scattering of anti-baryonic dark matter states (), annihilating baryons in the visible sector. The cosmic abundance of remnant and particles, with densities given by and, respectively, is the same as the baryon density in the universe today, and thus would have the same abundance ratio as, the baryon to photon () ratio. That is,

for a baryon density. Further details of this extension to SM is beyond the scope of this paper. From [8] (Equation (10)) we can relate the ratio of the density of dark matter (anti-baryonic) to baryonic matter in the universe to the ratio of the rest masses of the, and proton by

where and are, respectively, the and particle rest masses and is the proton rest mass and we assumed that. For the CGR model, (88) yields

which implies

This gives rest mass energies for the and particles of

For the model, (88) yields

which implies

This gives rest mass energies for the and particles [8] of

In both cases above we have tried to account for the constraint that, where is the electron mass, by restricting the range of values to be within. This may be only an approximate treatment, at best.

We now consider two SNe-Ia (SNe) light curve experiments [2] [3] . Common to both experiments is the stretch of the SNe light curve interval for each distant source when compared to the standard nearby (local) source. Because these studies rely on a model for what the light curve looks like in the rest frame of the source SNe, we will require the use of both kinds of cosmic time additions described above. The light curve time interval

from the distant SNe will be observed to have a time transformation which is a combination of cosmic time addition in the past combined with the time effect of the expansion of space, having the observed value which is given by (72),

In CSR this is the time interval that is observed from a source light curve or any time varying phenomenon at a cosmic time t in the past.

On the other hand, we can describe the light curve recorded by the observer in the frame at cosmic time t relative to our local observer in K at cosmic time 0. A light curve time duration of in K, from (71) for a cosmic time addition in the present, corresponds to the value in given by

If we make the assumption that

then combining (95) with (97) yields

It is evident that CSR, assuming (97), is consistent with the time dilation reports showing effects of cosmic aging equivalent with for redshift z. However, to offer a different perspective on cosmic time transformation, we will show plots of the ratio given by (95), instead of which was used in those reports.

For the SNe data from [2] the light curve agings are given as the light curve width w and the error in the width, obtained directly from [2] (Table 1) for the SCP high z SNe and from [2] (Table 3) for the Calán/Tololo low z SNe. Since the goal of the experiment was to normalize each light curve to a single standard light curve, we

will assume that the equivalent local rest frame time is which implies from (96) that. The reduced observed quantity is and we will assume using (98),

We will use (99) to acquire the redshift z and hence from the light curve rather than using the redshift from the host galaxy. The quantity we use is the ratio from (95),

The plotted data are shown in Figure 6. For data points with the redshift was set to. The plot shows the reduction of apparent light curve aging at higher redshift. This is the effect which would be seen in the observed light curve without scaling by the rest frame aging rate. The reduced chi-squared is for the data fitted to the cosmic aging rate, (74).

The next SNe data are from [3] . We take the aging rates from [3, Table 3], in the last column, unparenthesized. We compute

The errors come from [3] (Table 3), in the last column, parenthesized. The redshifts are computed from the aging rate data instead of from the given host galaxy values. The plotted data are shown in Figure 7. Again we note the reduction in the aging rate at higher redshifts. The reduced chi-squared for the data fitted to the cosmic aging rate is. In Figure 8 we show the combination of all the SNe aging data from the two reports. With 93 total data points the reduced chi-squared is.

The purpose of this section is to simulate quasar like light curve power spectra to compare with the report of observed low and high redshift quasar light curve power spectra [1] . The observed light curve power

spectra [1] (Figure 5, left-hand panel) were found to be identical within the experimental errors. Therefore, we will assume the low and high redshift light curves are identical in the observer frame K. In addition, it is assumed that the redshifts are of pure cosmological origin, with no components of gravitational redshifts or

Doppler shifts. In our simulation, the pseudo quasar light curve apparent magnitudes at epoch j is generated by the function

for each epoch, where, years and. The redshifts used are low and high and are from the quasar time dilation report [1] . For better resolution we used instead of the 28 yr which was used in the report. The Fourier power spectrum is determined from the magnitudes by [1] (Equation (1))

where j runs over equally spaced epochs of simulated data separated by time. Then the time transformations will take us from the origin of observer frame K to the quasar rest frame at cosmic time t. For CSR the sampling interval we use is defined by

where is given by (74) and

We divide by in (104) because we are going back in time to the quasar rest frame.

For our purposes, the flat space Friedmann-Lemaître-Robertson-Walker (FLRW) model is the CSR model with the cosmic aging function replaced by. For the flat space FLRW model we use for,

where we have again divided out the time transformation to get to the quasar rest frame.

For either model we use the fitting function defined by [1] , Equation (2)

where is the power, f is the frequency, is the redshift dependent frequency at maximum power and a and b are constants. We use the appropriate form of for the CSR or the FLRW models. We show light curve power spectrum plots of vs. for the light curves at low and high redshift.

Assuming both quasars have identical power spectra in the observer frame K, we obtain the power spectrum from (103) by setting the redshift which is given by. This is shown in Figure 9 with the fitting power function parameters, , where from [1] (Table 1, Observer frame Sample = z < 1, Index = −0.81) and. This can be compared with [1] (Figure 5, left-hand panel).

Next we show the light curve power spectra for the low and high redshift quasars, and, respectively, as it would be observed in their rest frame. The light curves are corrected by since we are obtaining the light curve back in time. We show the quasar power spectrum along with the fitting function, which has the same parameters as were used at the origin of K except for the frequency at maximum power which is given by

with from (74). This is plotted in Figure 10. The fitting function has the same parameters for both the low and high redshift spectra as were used in the spectrum at the origin of K except for.

To show what the power spectra might look like for a flat space FLRW observation we show the low and high redshift quasar light curve power spectra when the light curves are corrected for time dilation by, assuming a flat space cosmology. This is shown in Figure 11. This plot is similar to [1] (Figure 5, right-hand panel). The fitting function has the same parameters for both the low and high redshift spectra as were used at the origin of K except for the frequency at maximum power which is given by

In Figure 12 we show a contour plot of the cosmic aging ratio defined by

between two source fields, one at redshift z and the other at redshift. This is to demonstrate that it is possible to obtain similar aging rates (eg. within 10%) between two sources separated by large redshift. For the above low and high redshifts,.

For another demonstration of the cosmic aging ratio, we show in Figure 13 the power spectrum for each quasar in its rest frame where and. The aging ratio is.

In CGR the big bang occurred at velocity. At the recombination of protons and electrons in the baryon-photon plasma, when the photons decoupled to form the CMB radiation field, the velocity was,

which is related [4] to the time of recombination by. Applying (58), we have

where is the cosmological redshift at recombination. The coordinate distance to the big bang is given by (23) with,

Likewise, the coordinate distance to the recombination epoch with, is given by

We construct a simple model to determine the size of the sound horizon [16] [17] for the longest sound wave, which generates the first acoustic peak. If is the radius of the sphere of expanding plasma, is the average expansion velocity between the big bang and the recombination epoch and is the speed of the longest sound wave in the plasma, then, by proportion of velocities, is the radius of the sphere containing the longest wave. Assuming that the wave travels along a great circle path of the sphere, the size of the sound horizon is given by

Defining, which is the difference of (112) and (113), then from (114), the size of the sound horizon at recombination is given by

The angle of the sound horizon at recombination is given by

where the angular diameter distance is given by [10]

where is the luminosity distance of the recombination epoch. Substituting from (111)-(115) and (117) into (116) and simplifying we obtain

The CMB radiation escaped the matter sphere and expanded to fill all space. The size of the sound horizon in the CMB on today’s sky is obtained by applying (53) with,

Then, the angle of the sound horizon in the CMB radiation field on today's sky is given by

The multipole l of the first acoustic peak [16] recorded in the CMB radiation field is proportional to the inverse of (120),

Substituting and into the above equations we obtain a value of

which is in good agreement with observation [18] [19] and,

The size of the sound horizon on today’s sky is, which is 1/5 the value of the standard model.

Let us review briefly some aspects of the Carmeli five dimensional brane world cosmological model.

In this paper, we used the linearized approximation of the 5-D Cosmological General Relativity as developed by Carmeli. A flat space CSR model was derived in a general way from the curved space CGR model. The CGR luminosity distance relation was applied to SCP Union 2.1 SNe-Ia distance data up to redshift combined with GRB distance data up to redshift. Utilizing the reduced method in the data analysis, it was shown that the CGR model with a best fit mass density of performed as well as the flat space model with an apriori mass density of. Regarding the hypothetical X particle

constituents, the CGR model determines rest mass energies for the and particles of . We also found that CSR could confirm the model result of time dilation of in SNe-Ia light curves. We also showed how the cosmic aging function produced a null effect of time dilation in simulated light curve power spectra between two groups of hypothetical QSO’s separated by a redshift. Finally, we gave a model for obtaining the first acoustic peak of the CMB anisotropy, deriving a multipole, in good agreement with observation, and with an angle on the sky of.

Firmin J. Oliveira, (2016) Cosmic Time Transformations in Cosmological Relativity. Journal of High Energy Physics, Gravitation and Cosmology,02,253-279. doi: 10.4236/jhepgc.2016.22022