_{1}

^{*}

Cosmological redshift is commonly attributed to the continuous expansion of the universe starting from the Big-Bang. However, expansion models require simplifying assumptions and multiple parameters to get acceptable fit to the observed data. Here we consider the redshift to be a hybrid of two effects: recession of distant galaxies due to expansion of the universe, and resistance to light propagation due to cosmic drag. The weight factor determining the contribution of the two effects is the only parameter that is needed to fit the observed data. The cosmic drag considered phenomenologically yields mass of the universe
≈ 2 × 10
^{53} kg. This implicitly suggests that the mass of the whole universe is causing the cosmic drag. The databases of extragalactic objects containing redshift
*z* and distance modulus
μ of galaxies up to
*z* = 8.26 resulted in an excellent fit to the model. Also, the weight factor
*w _{D}* for expansion effect contribution to
μ obtained from the data sets containing progressively higher values of
μ can be nicely fitted with
.

Alternative explanations to Doppler effect, or expansion of the universe, for observed redshift of luminous objects in distant galaxies, such as tired light models, have never been taken seriously since it was first offered by Zwicky in 1929 [

The mechanism that leads to the loss of energy in tired light models has not been made clear in most of the studies although Compton scattering, or like models, have been cursorily suggested. The most used form of the tired light approach takes an exponential increase in photon wavelength with distance traveled:

λ o = λ e e d R o , (1)

where λ o is the observed wavelength of the photon at distance d from the source of emission, λ e is the wavelength of the photon at the source of light and R o is a constant that characterises the effect of the cause of the increase in wavelength whatever that may be.

The focus here is to derive Equation (1) from a simple model of resistance of the fields in space to the propagation of photons (and possibly other particles), similar to that of the propagation of a particle through a resistive field of a fluid in fluid dynamics

In fluid dynamics, the particle ceases to accelerate when the applied force on a particle F equals fluid’s resistance or drag:

F = 1 2 ρ v 2 A C d . (2)

Here ρ is the density of the fluid through which the particle is propagating, v is the particle velocity, A is the particle area and C d is the fluids drag coefficient. Now this force F may also be written as − d E d x where dE is the energy used up in moving the particle a distance dx in the fluid.

− d E d x = 1 2 ρ v 2 A C d . (3)

Inspired by this equation, in our phenomenological cosmic drag model for a photon traveling through space, we write as follows:

E = h ν , with h as Planck’s constant and ν as photon frequency,

A C d is assumed to be proportional to the energy E,

ρ is a constant related to the entity causing the drag,

v = c , the speed of light.

We may then write:

− d ( h ν ) d x = ( h ν ) c 2 κ . (4)

Here κ is a constant that captures 1 / 2 ρ and the proportionality constant that relates E to A C d , thus representing the resistive properties of the cosmic drag fields on the photon. Integrating Equation (4) over distance d from the photon emission point to the photon observation point, we have:

ln ( ν e ν o ) = c 2 d κ , (5)

or

ln ( λ o λ e ) = c 2 d κ , (6)

or

λ o λ e = e c 2 d κ . (7)

Here, ν e and ν o are respectively the emitted and observed photon frequencies and λ ν = c . Now, since the redshift is defined as z = λ o λ e – 1 , we may write Equation (7):

z + 1 = e c 2 d κ , (8)

or

ln ( 1 + z ) = c 2 d κ (9)

The constant κ can be determined from the small redshift limit of Equation (9) by appealing to the Hubble law. The law may be written for small z as c z = H o d , where H o is Hubble constant and d is the distance of a galaxy with small redshift. This allows us to write for small values of z

ln ( 1 + z ) ≈ z = c 3 z H o κ , (10)

or

κ = c 3 H o , and (11)

d = ( c H o ) ln ( 1 + z ) . (12)

Taking c = 3 × 10 8 m / s and H o = 2 × 10 − 18 / s (= 70 km/s/Mpc), we get κ = 1.35 × 10 43 m 3 ⋅ s − 2 . Before we proceed further let us see if this constant has some cosmological meaning.

Looking at Newton’s gravitational constant G N , we notice that its dimensions are m^{3}×k^{−1}×s^{−2} with a value of 6.674 × 10^{−11}. Thus on the dimensional ground,

κ = M x G N , (13)

where M x is an unknown mass factor related to the propagation of light in the universe. This yields

M x = c 3 H o G N = 2 × 10 53 kg . (14)

We can readily recognize this as the mass of the observable universe (Hoyle-Carvalho formula [

Substituting κ from Equation (13) into Equation (4), we have,

− d ( h ν ) d x = ( h ν ) c 2 M x G N . (15)

This equation shows that the drag on the photon depends on the mass of the observable universe and thus it is a manifestation of Mach’s Principle [

We will now proceed to fit the observed redshift data using the Doppler effect (including expansion effect) based model and the Mach effect based model proposed here, to explore if one or the other gives a better fit, or perhaps both the effects are partially accountable for the observed redshift. The model we chose for the first type is that recently developed analytically by Mostaghel [

1) A set of 557 SNe data with redshifts from 0.0152 ≤ z ≤ 1.4 as compiled in the 2010 in the Union2 database [

2) A set of 394 extragalactic distances to 349 galaxies at redshifts 0.133 ≤ z ≤ 6.6 as reported in 2008 NASA/IPAC’s NED-4D database [

3) Data for three most distant recently confirmed galaxies [

The distance modulus μ and the redshift z are represented by Mostaghel [

μ = 5 log [ R o ( 1 − a ) K ( z ) ] + 25 , (16)

where a is the scale factor, R o = c / H o is in mega parsecs, and K ( z ) includes K-correction that corrects observation data for source luminosity, instrumental factors, and other factors. With 1 − a = z / ( 1 + z ) and K ( z ) = ( 1 + z ) b ,

μ = 5 log [ R o ( z / ( 1 + z ) ) ( 1 + z ) b ] + 25 . (17)

Mostaghel fitted 1^{st} set of data in Equation (17), and found b = 5 / 3 . This equation was used to fit all the three sets of data showing a reasonably good fit. (It should be mentioned that we found for all the three data sets a better fit is obtained by using b = 1.487 and not by using b = 5 / 3 ). He used his analytically derived value of H o = 69.05398 km/s/Mpc in Equation (17) as he found it to be very close to the average of the most recently reported value of the Hubble constant. He found z − μ plots using Equation (17) were in good agreement with CDM model fit with the same data using the scale factor given by equation

1 / a ( z ) = ∫ 0 z [ Ω m ( 1 + z ′ ) + Ω A ( 1 + z ′ ) − 2 + Ω r ( 1 + z ′ ) 2 + Ω k ] − 1 2 d z ′ + 1 , (18)

with H o = 68.45 ± 0.96 , Ω A = 0.703 ± 0.012 , Ω m 0 = 0.297 ± 0.012 , Ω r 0 = 0 and Ω k = 0 . We there for used Equation (18) as representing the expansion model, i.e. the Doppler effect model.

Based on Equation (12) for Mach effect model, distance modulus may be written as

μ = 5 log [ R o ln ( 1 + z ) K ′ ( z ) ] + 25 , (19)

with K ′ ( z ) = ( 1 + z ) d is correction factor for Mach Effect and d is determined by fitting the observational data.

The observational data we chose for our study is only slightly different from Mostaghel’s data discussed above. We took (a) a set of 580 SNe data with redshifts from 0.015 ≤ z ≤ 1.414 as compiled in the 2010 in the Union2 database [

As the redshift may be partly due to Doppler effect and partly due to Mach effect, we also considered fitting the observed data with weight factors given to Equations (17) and (19) and determining the weight factors with nonlinear regression analysis. Thus, we may write

μ = w 5 log [ R o ln ( 1 + z ) ( 1 + z ) d ] + ( 1 − w ) 5 log [ R o ( z 1 + z ) ( 1 + z ) b ] + 25 , (20)

where w is the weight factor given to Equation (19) and ( 1 − w ) to Equation (17); two weight factors must add up to 1 and 0 ≤ w ≤ 1 . Parameters b and d we tried in Equation (20) for determining w are as follows: 1) determined from fitting data set (a), that is b = 1.671 and d = 1.194 ; 2) determined from fitting the combined dataset (a) and (b), that is b = 1.487 and d = 1.042 ; and 3)

b = 2 and d = 1 . While parameters b and d may also be determined along with by fitting Equation (20) directly to the observed data, their variance becomes very high due to significant scatter in observed data. The fitted curves for the three cases are plotted in

Model reference | Data base | Data points | Model parameters | 95% Confidence | Goodness of fit | ||||
---|---|---|---|---|---|---|---|---|---|

Label | Value | SSE | R-Squared | RMSE | |||||

Doppler | a | 580 | b | 1.671 | 1.643 | 1.699 | 41.78 | 0.9929 | 0.2685 |

Mach | a | 580 | d | 1.194 | 1.166 | 1.222 | 41.53 | 0.993 | 0.2678 |

Doppler | a + b | 962 | b | 1.487 | 1.468 | 1.506 | 308.1 | 0.981 | 0.5666 |

Mach | a + b | 962 | d | 1.042 | 1.023 | 1.06 | 302.2 | 0.9814 | 0.5611 |

Case 1 | a + b | 962 | w | 1 | fixed at bound | 381.9 | 0.9765 | 0.6304 | |

Case 2 | a + b | 962 | w | 1 | fixed at bound | 302.2 | 0.9814 | 0.5608 | |

Case 3 | a + b | 962 | w | 0.9276 | 0.8936 | 0.9615 | 302.6 | 0.9814 | 0.5614 |

The weight factor appears to strongly favour Mach effect; w ≈ 1 . However due to logarithmic dependence of μ on z, w is also strongly dependent on parameters b and d of the K ( z ) and K ′ ( z ) factors, which in turn heavily depends on the K-correction. Here we are assuming that both the effects determine μ and z. Then, if we use the equation that only represent one effect, the exponent of ( 1 + z ) x , with x = b or d, has to take care of not only the K-correction, etc., but also for the other effect. Since d comes out to be up to 20% greater than 1, while b comes out to be up to 25% less than 2 (first four rows of

μ = 5 ( 1 − w D ) log [ R o ln ( 1 + z ) ( 1 + z ) ] + 5 w D log [ R o z ( 1 + z ) ] + 25 , (21)

where w D = ( 1 − w ) is now the Doppler effect weight factor. Sixteen data sets were created with progressively increasing value of μ ; say for μ = 40 all the data up to μ = 40 was included. For each data set, w D was determined by fitting the data using Equation (21). Resulting 16 data points ( μ , w D ) were then fitted using a Gaussian function with the constraint that the factor w D satisfy the condition 0 ≤ w D ≤ 1 . The plot is shown in

One problem with this plot we noticed is that the constraint w D = 0 was hit 8 times. This suggests that w D has a tendency to go negative. When we removed the constraint on w D , we got the data points that fitted beautifully a two term sine function (

w D ( μ ) = a 1 sin ( b 1 μ + c 1 ) + a 2 sin ( b 2 μ + c 2 ) , (22)

with a 1 = 0.198 , b 1 = 0.4159 , c 1 = 2.049 ; a 2 = 0.2418 , b 1 = 0.6768 , c 2 = 5.15 . This amounts to the Doppler effect contribution in Equation (21) to be negative in some regions and positive in others.

We may interpret the positive w D as indicative of the expansion of the universe and negative as contraction. As per

It should be mentioned that the parameter b and d in

The extragalactic redshift has been shown to be due partly to the Doppler effect (expansion of the universe) and partly due to Mach effect by analysing up to date data available from Union2 and NASA/NED data bases. The model resulting in Mach effect yields mass of the observable universe as

We wish to thank Professor Naser Mostaghel for providing the SNe and Union2 observational data used in his research paper [

Gupta, R.P. (2018) Mass of the Universe and the Redshift. International Journal of Astronomy and Astrophysics, 8, 68-78. https://doi.org/10.4236/ijaa.2018.81005