Cosmology/Dichotomous

Introduction
The so-called dichotomous cosmology also referred to as the luminous universe, was introduced by Y. Heymann in 2014 as inspired by tired-light theory, on the basis of the exponential law which derivative is equal to itself. It describes a universe where the material world is static and the luminous world expanding. This cosmology makes it possible to reconcile the static universe of Einstein with observations of the expanding universe. Specifically, the theory is reported to conform with the following observations: the relationship between luminosity distance and redshift of supernovae, the stretching of supernova light curves by a factor $$(1+z)$$, and the factor $$(1+z)^4$$ for the radiation energy density inferred from the Cosmic Microwave Background Radiation.

Cosmological distance measurements
Distance measurements in the dichotomous cosmology may be derived in two ways: from the tired-light paradigm, and from the expanding metrics. Both derivations lead to the same equations.

Derivation from tired-light paradigm
When a photon loses energy during its travel in space, the wavelength of light is stretched, and because the number of light wave cycles is conserved, an expansion of the luminous world is produced. As a consequence of this stretching of light, the velocity of the light wavefront increases during its travels. According to special relativity, the speed of light is invariable. Hence, in order to maintain the light wavefront at the speed of light, the model introduces a time contraction between the source and the observer.



Redshift and photon energy
Considering that photons lose energy as light gets stretched, the following equations are obtained:

where $$E(z)$$ is the photon energy when emitted, $$E_0$$ the photon energy at reception and $$z$$ the redshift.

A simple law of decay of the photon energy is considered:

$$\frac{\dot{E}}{E} = -H_0$$.

Therefore

and

where t is the time which is equal to zero at time of observation, and T the light travel time from the observer.

A set of two transformations is considered: first a time variable light wavefront to accommodate for the expansion of the luminous world, and second a time contraction in the arrow of time to maintain the light wavefront at the speed of light.

Light wavefront with respect to the source
The light wavefront velocity is

$$v(t) = c \frac{E_{emit}}{E(t)}$$.

To maintain the light wavefront at the speed of light, the following time contraction is applied:

$$\frac{\delta t^\prime}{\delta t} = \frac{E_{emit}}{E(t)}$$.

Hence, the light travel time with respect to source is

$$T^\prime = \int_{-T}^{0}\frac{\delta t^\prime}{\delta t} dt = \int_{-T}^{0}\frac{E_{emit}}{E(t)} dt$$.

Introducing (2) in the previous equation and integrating, we get:

$$T^\prime= \frac{E_{emit}}{E_0}\frac{1}{H_0} \left(1- \exp(-H_0 T) \right)$$.

By substitution of (3) in the previous equation:

$$T^\prime= \frac{E_{emit}}{E_0}\frac{1}{H_0} \left(1- \frac{E_0}{E_{emit}} \right)$$.

Introducing (1) in the previous equation, we get

which is the light travel time measurement for the luminosity distance.

Light wavefront with respect to the observer
The light wavefront velocity is

$$v(t) = c \frac{E_0}{E(t)}$$.

To maintain the light wavefront at the speed of light, the following time contraction is applied:

$$\frac{\delta t_0}{\delta t} = \frac{E_0}{E(t)}$$.

The light travel time with respect to the observer is:

$$T_0 = \int_{-T}^{0} \frac{\delta t_0}{\delta t} dt = \int_{-T}^{0} \frac{E_0}{E(t)} dt$$.

Introducing (2) in the previous equation and integrating:

$$T_0 = \frac{1}{H_0} \left(1-\exp(-H_0 T) \right)$$.

Introducing (3) in the previous equation:

$$T_0 = \frac{1}{H_0} \left(1-\frac{E_0}{E_{emit}} \right)$$.

Introducing (1) in the previous equation, we get

which is the light travel time measurement for the actual distance.

Luminosity distance
The luminosity distance is the distance measured from the luminosity of supernovae. Supernovae Ia are considered standard candles, meaning that they all have the same absolute brightness when they explode. From their apparent brightness, we can deduce the luminosity distance, because the brightness diminishes proportionally to the inverse of the distance squared. The formula used to compute the luminosity distance is the distance modulus equation.



For the calculation of the luminosity distance the light travel time with respect to the emission point $$T^\prime$$ derived earlier is used.

The luminosity distance $$r_L$$ is expressed as follows:

$$\frac{dr_L}{dt^\prime} = c$$. By integrating between 0 and $$T^\prime$$, we get $$r_L = c T^\prime$$.

Introducing (4) in the previous equation yields :

To use this equation with supernova data, the redshift adjusted distance modulus equation is obtained based on photon flux. The redshift adjusted distance modulus equation is as follows:

$$m-M = -5 + 5 \log(r_L) + 2.5 \log(1+z)$$,

with m the apparent magnitude and M the absolute magnitude.

A linear relationship for the luminosity distance versus redshift was obtained, with the inverse of the slope being the Hubble constant. The Hubble constant obtained from the regression of the luminosity distance versus the redshift is 63 km/s/Mpc, or 0.064 per Gyr (billion years).

Derivation from expanding metrics
In the dichotomous cosmology, the luminous world is expanding; therefore, we can derive the distance measurements using expanding metrics.

Luminosity distance
By considering a photon travelling away from the center of a supernova, the luminosity distance is calculated as follows:

$$\frac{dr_L}{dt}= c + H_0 r_L$$.

By integrating this equation between 0 and T, we get:

Because $$\frac{da}{dt}= H_0 a$$, we get: $$dt = \frac{da}{H_0 a}$$, where a is the scale factor. In addition, the relationship between the scale factor and the redshift is given by the cosmological redshift equation $$(1+z)= \frac{1}{a}$$, where the scale factor is equal to one at present time.

Hence, the light travel time versus redshift is as follows:

Eqs. (7) and (8) yield:

which is identical to (6).

Euclidean distance
A measurement of the distance is obtained by calculating the corresponding distance if there were no expansion, which we call the Euclidean distance. Let us introduce $$y$$ to this distance measurement. By considering a photon moving towards the observer, we get:

$$\frac{dy}{dt}= -c + H_0 y$$.

By setting time zero at a reference Tb in the past, we get $$t=T_b-T$$; therefore, $$dt=-dT$$. Hence:

$$\frac{dy}{dT}= c - H_0 y$$, with boundary conditions $$y(T=0)=0$$.

Integrating this equation between 0 and T, we get:

By substitution of (8) into (10), we get:

which is identical to (5) with $$T_0=\frac{y}{c}$$.

Etherington's distance-duality equation
The Etherington's distance-duality equation is the relationship between the luminosity distance of standard candles and the angular-diameter distance. The equation is as follows:

$$r_L = d_A (1+z)^2$$,

where $$r_L$$ is the luminosity distance and $$d_A$$ is the angular-diameter distance.

The angular-diameter distance of an object is defined in terms of the object's actual size, $$x$$, and $$\theta$$ the angular size of the object as viewed from earth: $$ d_A= \frac{x}{\theta}. $$

Because of the expansion of the luminous world, the apparent size of celestial objects is stretched by a factor $$(1+z)$$, and the apparent angular size is increased by the same factor. Hence, the relationship between the actual distance $$y$$ and the angular-diameter distance is as follows:

$$y = d_A (1+z)$$.

Therefore, we get:

$$ r_L =(1+z) y$$.

This is the same relationship we obtain from (6) and (11).

Although the Etherington's reciprocity theorem is often considered to be peculiar to cosmological models based on Riemannian geometry, this relationship follows naturally from the dichotomous cosmology. The Etherington's reciprocity theorem has been verified using astronomical observations based on X-ray surface brightness and the Sunyaev-Zel'dovich effect of galaxy clusters.

The cosmological test
A cosmological test based on the zCOSMOS observations carried out using the Very Large Telescope at the ESO Paranal Observatory is established to test the dichotomous cosmology against a specific class of expanding universes: universes with a Hubble parameter which does not vary over time.

The rationale of the test is to slice the zCosmos galactic survey into small redshift buckets. For each redshift bucket, we compute the number of galaxies in the bucket divided by the volume of the bucket, which gives the galactic density of the bucket. Using this procedure, a curve of the galactic density versus light travel time is obtained. Then the theoretical galactic density curve of the cosmology is obtained by simulation by generating galaxies for each redshift bucket with a uniform distribution, and computing the number of visible galaxies (those not covered by foreground galaxies) using an average galactic radius. Finally, by comparing the galactic density curve of the simulation with that of the survey we can accept or reject a cosmology. The source code for the simulation is available on the Codeproject website.

This test is favourable for the so-called dichotomous cosmology while it rejects the expanding universe classes considered.