Creation Moment 7/30/2024 - A New Cosmology SERIES: Distances in the FLRW Metric

 I am the LORD that maketh all things; 

that stretcheth forth the heavens alone; 

that spreadeth abroad the earth by Myself; 

That frustrateth the tokens of the liars, 

and maketh diviners mad;

 that turneth wise men backward, 

and maketh their knowledge foolish; 

Isaiah 44:24,25

"In an expanding space, the perceived brightness of a distant standard candle is no longer simply inversely proportional to the square of the distance. 

And the angular diameter is no longer inversely proportional to distance. 

Both of these are good approximations for nearby objects, but fail for objects with significant redshifts due to the curvature of spacetime.

To deal with these general relativistic effects, astronomers use several different definitions of distance, and convert between them as needed. 

--The comoving distance uses coordinates that expand with space, maintaining number density within a comoving volume. 

--Luminosity distance is the distance an object would be at if its brightness diminished as the inverse square of distance.

So, by definition, an object’s apparent brightness diminishes as the inverse square of its luminosity distance. 

--Finally, the angular diameter distance is the distance at which an object would be for its angular diameter to be inversely proportional to distance. So, an object’s angular diameter (θ) is simply its true diameter (s) divided by its angular diameter distance (DA).

𝜃=𝑠𝐷𝐴

These three functions of redshift each depend on three cosmological parameters: the Hubble constant (H0), matter density of the universe relative to critical density (ΩM), and dark energy density relative to critical density (ΩΛ). 

We will here assume the standard “flat” universe in which the sum of matter density and dark energy density equals one: ΩM + ΩΛ = 1. And going forward we will assume the standard values of the cosmological parameters: ΩM = 0.3; ΩΛ = 0.7; H0 = 70 km/s/Mpc. Only these values are consistent with observations of the nearby universe under the assumption of the FLRW metric and homogeneity (Choudhury and Padmanabhan 2005: Lisle 2016). 

This is called the Lambda-Cold-Dark-Matter (ΛCDM) model, or simply the standard model. In a “flat” universe, the comoving distance (DC), luminosity distance (DL), and angular distance (DA) are related to redshift (z) by the following (Hogg 2000):

𝐷𝑐(𝑧)=𝑐𝐻0∫0𝑧(𝑑𝑧′/1+Ω𝑀((1+𝑧′)3−1))

𝐷𝐿(𝑧)=𝑐(1+𝑧)𝐻0∫0𝑧(𝑑𝑧′/1+Ω𝑀((1+𝑧′)3−1))

𝐷𝐴(𝑧)=𝑐𝐻0(1+𝑧)∫0𝑧(𝑑𝑧′/1+Ω𝑀((1+𝑧′)3−1))

Notice that all these distances converge at low z. This is because the spatial geometry of a ΛCDM universe is nearly Euclidian for very short distances. 

At higher distances, the luminosity distance is always larger than the comoving distance, indicating that objects will appear fainter than they would if stationary in a non-expanding universe at that distance. The angular diameter distance is particularly interesting because it reaches a maximum at z ≈ 1.6. Since the angular diameter distance is the inverse of the angular diameter of an object at redshift z, this means that an object will continue to look smaller with increasing distance until it reaches redshift 1.6. Beyond that distance, the object will start to look larger as its distance increases.

--Thus, all other things being equal, galaxies should appear smallest at a distance corresponding to redshift 1.6 according to the standard model. 

--Beyond that distance, galaxies should appear larger as their redshift increases. However, this effect is simply not seen in any Hubble Space Telescope (HST) or JWST images. Indeed, galaxies continue to appear smaller with increasing distance as if the large-scale spatial geometry of the universe were purely Euclidian as will be shown below. This is what we would expect if the Hubble law were due to Doppler shifts of galaxies moving through a non-expanding space.

So, we now consider the possibility that galactic redshifts are caused purely by the Doppler effect due to their motion through non-

expanding space. Furthermore, let’s assume for the sake of hypothesis that the Hubble law is truly linear with respect to velocity even at high redshifts. In this case, the Hubble flow is not due to expansion of space itself, but merely reflects the average recessional velocity of galaxies at a given distance. Then, neglecting peculiar velocities, the distance to any galaxy is simply its velocity (as derived from redshift) divided by the Hubble constant which we take to be 70 km/s/Mpc. We will call this the Doppler model.

The recessional velocity (v) of an object with a Doppler induced redshift (z) is given by the following formula (Lisle 2018)2:

𝑣=𝑐(𝑧+1)2−1(𝑧+1)2+1

The Doppler model therefore does not predict the magnification effect that is required in the ΛCDM model at redshifts greater than 1.6. We have also plotted the distance predicted by the tired light model. This would also be the angular diameter distance in the tired light model, since it presupposes that galaxies are essentially stationary (aside from their small peculiar velocities) in a non-expanding space.

In the Doppler model, luminosities do not diminish as 1/r2. This is due to relativistic beaming. Beaming is caused by Lorentz aberration (isotropic radiation in the rest frame will not be isotropic in a moving frame), and the time dilation. The formula for luminosity reduction (LR) by beaming at velocity (v) for an object moving directly away from the observer is (Rybicki and Lightman 1979):

𝐿𝑅=(1−𝑣/𝑐1+𝑣/𝑐)3

This can also be expressed in terms of redshift by substitution of v from equation:

𝐿𝑅=1/(𝑧+1)3

Thus, we can compute the luminosity distance according to the Doppler model:

𝐷𝐿𝐷𝑜𝑝=𝐷𝐷𝐿𝑅=𝐷𝐷(𝑧+1)32

Since this curve is higher than the luminosity distance for ΛCDM, the Doppler model predicts that galaxies will appear slightly fainter at a given redshift than the prediction of the standard model." AIG