Another Look at Taking Another Look at the Universe

In Mathematics for Taking Another Look at the Universe, I presented this chart (discussed further in Apparent Hubble Parameter Value), which plots the value of the Hubble parameter:

I should be explicit about how this chart was generated.  First, I should explain the calculations used in generating the chart discussed in Mathematics for Taking Another Look at the Universe, first introduced in Taking Another Look at the Universe:

This chart illustrates the proper distances (x') of a photon over time from the location of an observer at (x,t)=(0,0), in a FUGE universe of age t₀=13,000 million years, given that the photon is observed by that observer.

To generate the chart, I had three columns in a spreadsheet (the red line is actually just a line overlaid over the chart).  There was one for the x-axis, from 0 to 13,800 (millions of years) at intervals of 50, one for the blue curve (x'=(13800-x).x/13800)) and another for the grey line, from 13,800 (millions of light years) to 0 also at intervals of 50.

The blue curve plots every event through which a photon emitted very early in the FUGE universe’s chronology must pass in order to be observed.  That path is one through spacetime and has a spacetime length of 15,840 million (light) years.  This should be compared with a hypothetical static universe in which a photon reaching us after time t would have travelled across a distance of ct, so the magnitude of the spacetime distance would have been √2 times that.  Such a path could be normalised by merely multiplying by (distance/spacetime distance).

To generate the Hubble parameter chart, I replicated the first two of the three columns discussed above with the x value multiplied by 1.000001. Then I created a column each for Δt (s), Δx (km), x (Mpc), x_ST (Mpc) and Δx_ST (km).

The first column, Δt, contained only the equivalent of 0.000001 million years (4.3548×10¹¹ seconds) in every row.  The second, Δx, had the difference in the values of x before and after multiplication by 1.000001, expressed in km.  The third column merely converted the original values of x into Mpc.

The fourth and fifth columns, x_ST and Δx_ST, need some explanation.  They relate to the spacetime pathlength of a photon through all observable events, as represented above, that has a length of 15,800 million (light) years.  So, x_ST(t) is the length between x'(0) and x'(t), while Δx_ST is the delta between the x_ST(t₁) and x_ST(t₂) where Δt=|t₂-t₁|, where both are normalised in a similar way to as discussed above, specifically after division by 15,840 and multiplication by 13,800.

The Hubble parameter chart has two plots, a blue one and an orange one. The flat blue line results from the equations H₀=Δx/Δt/x=Δx_ST/Δt/x_ST=70.855 km/s/Mpc.  The orange curve plots apparent a hypothetical “apparent Hubble parameter” (H_App) from the equation H_App=Δx_ST/Δt/x, on which H_App varies between 70.855 and 87.143 km/s/Mpc.  (Note that if Δx_ST is not normalised, the H_App value ranges from 78.784 to 100.022 km/s/Mpc.  This is interesting, given that an H₀ magnitude of 100 appeared frequently before 2013, but that value should not be taken as meaningful, since 100 could be more realistically thought of as an order of magnitude, 10², rather than 1.00×10².)

In Apparent Hubble Parameter Value, I wrote “I cannot say with any confidence that the apparent acceleration of the universe (using recent measurements of nearer galaxies) is due to an artefact of measurement related to blending values of x=ct and x'.”  This still stands but note that, per the above, the use of x' not entirely accurate.  It’s really a blending of x=ct and x_ST.

Note that while I can’t say with confidence that a blending of x=ct and x_ST leads to apparent recent acceleration of the universe, because the evidence doesn't appear to exist, I can at least point to a mechanism by which measurement leads to it, as an artefact, rather than being a real thing.

Note also that what we have effectively been discussing above is the effect of recession on the location of an event.  What has not been discussed is how we, as observers, can determine speed of recession.  I’ll address that in the next post.