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Distance Calculation

In the Face Book group:
My post:

Astronomers have a small number of techniques to measure distances outside our galaxy.
The first two rely on a known luminosity curve for the object: either a Cepheid variable or a supernova type 1A. If the observed brightness is reduced then the distance can be deduced from that dimming.
The third technique relies on assumptions that could be debated. The spectrum  in the light from a distant object can be shifted. This is like the doppler effect in sound but it is  called a redshift for light, where a shift toward blue indicates an approach while a shift toward indicates recession. That is straightfoward. There are several problems with such a simple rule. 1) sometimes the red shift is faster than the speed of light which supposed to be impossible. 2) large redshifts are often quantized, meaning the observed shift is measured in increments. With random velocities and directions this should not be possible. 3) Sometimes a pair of objects will have the same large redshift, typically on opposite sides of a central galaxy with a lower redshift . Again with random directions or velocities this symmetry should be extremely unlikely. When plotting these observations, there will be concentric rings of objects around the earth with these quantized velocities so  the earth is the apparent center of the universe. This concept was abandoned years ago but red shifts brought it back.
Halton Arp offered a simple solution. Part of the large red shift is intrinsic meaning the spectrum was shifted at the moment it was emitted. This intrinsic component is quantized so it is unrelated to velocity and it could even imply a velocity more than the speed of light. This proposal seems to solve the apparent problems with observed redshifts.

According to wikipedia:

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Redshift quantization is a fringe topic with no support from mainstream astronomers in recent times. Although there are a handful of published articles in the last decade in support of quantization, those views are rejected by the rest of the field.
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Cosmologists just ignored these observations and went on to a new concept: an expanding universe. This concept allowed speeds faster than light and also allowed red shift to define distance. A shift in spectrum is a result of only relative velocity. The is no distance component present. Cosmologists have added the new concepts of cosmic time, comoving distance, proper distance, and cosmological redshift.
From wikipedia:

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Cosmic time (also known as time since the big bang) is the time coordinate commonly used in the Big Bang models of physical cosmology. It is defined for homogeneous, expanding universes as follows: Choose a time coordinate so that the universe has the same density everywhere at each moment in time (the fact that this is possible means that the universe is, by definition, homogeneous). Measure the passage of time using clocks moving with the Hubble flow. Choose the big bang singularity as the origin of the time coordinate.

Note: I posted the question in the last few days whether the universe is homogeneous.

From wikipedia:
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Proper distance roughly corresponds to where a distant object would be at a specific moment of cosmological time, which can change over time due to the expansion of the universe.

. The comoving spatial coordinates tell where an event occurs while cosmological time tells when an event occurs. Together, they form a complete coordinate system, giving both the location and time of an event.

Space in comoving coordinates is usually referred to as being "static", as most bodies on the scale of galaxies or larger are approximately comoving, and comoving bodies have static, unchanging comoving coordinates. So for a given pair of comoving galaxies, while the proper distance between them would have been smaller in the past and will become larger in the future due to the expansion of space, the comoving distance between them remains constant at all times.

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Cosmological redshift is the measured red shift but adjusted for the expansion of the universe. This calculation uses the Lemaitre equation, see below a statement about the FLRW metric (the L is Lemaitre). The cosmological redshift is calculated based on an expanding universe and then the cosmological redshifts confirm the universe is expanding.
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The expanding Universe has an increasing scale factor which explains how constant comoving distances are reconciled with proper distances that increase with time.
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The goal of these concepts is cosmologists wish to define the distance this red shift object has moved since the big bang given only its current observed velocity. Then with the assumption the big bang was about 13.8 billion years ago a distance to the object can be calculated relative to that 13.8 billion LY benchmark.

From wikipedia on the most distant object:

'With an estimated light-travel distance of about 13.4 billion light-years (and a proper distance of approximately 32 billion light-years (9.8 billion parsecs) from Earth due to the Universe's expansion since the light we now observe left it about 13.4 billion years ago), astronomers announced [GN-z11] as the most distant astronomical galaxy known.
From wikipedia:
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The Friedmann–Lemaître–Robertson–Walker (FLRW) metric is an exact solution of Einstein's field equations of general relativity; it describes a homogeneous, isotropic, expanding (or otherwise, contracting) universe that is path-connected, but not necessarily simply connected. The general form of the metric follows from the geometric properties of homogeneity and isotropy; Einstein's field equations are only needed to derive the scale factor of the universe as a function of time. Depending on geographical or historical preferences, the set of the four scientists — Alexander Friedmann, Georges Lemaître, Howard P. Robertson and Arthur Geoffrey Walker are customarily grouped as Friedmann or Friedmann–Robertson–Walker (FRW) or Robertson–Walker (RW) or Friedmann–Lemaître (FL). This model is sometimes called the Standard Model of modern cosmology, although such a description is also associated with the further developed Lambda-CDM model. The FLRW model was developed independently by the named authors in the 1920s and 1930s.
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There is always  the assumption the universe is homogeneous and isotropic.

The FLRW model was developed over 70 years ago.

For the Lambda-CDM model:
Much more precise spacecraft measurements of the microwave background from WMAP in 2003 – 2010 and Planck in 2013 - 2015 have continued to support the model and pin down the parameter values, most of which are now constrained below 1 percent uncertainty.

In another recent post (about a day ago) I pointed out there are known problems with those microwave measurements.

After reviewing the basis for how a distance is calculated for large red shifts, ignoring a possible intrinsic redshift, with the persistent assumption of a homogeneous and isotropic universe.  I wonder how often cosmologists review all these assumptions in their models, .

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