Approximating function to The Hubble Diagram to Redshift > 6 from 69 Gamma-Ray Bursts

[Hannu Poropudas]

Approximating function to The Hubble Diagram to Redshift > 6 from 69 Gamma-Ray Bursts

Distance moduli
u as function of redshift z.

u = m - M = 5* log (d_L) + 25,

(d_L in units of megaparsecs),

d_L = luminosity distance,

u = distance moduli,
m = apparent magnitude,
M = absolute magnitude.


I made following approximative function of distance moduli u (magnitudes)
as function of redshift z of 69 GRBs.

GRB data ( 0.17 < z < 6.60 ):
----------------------------

u = 2.5649*ln(z) + 44.159 ,

R^2 = 0.9011 , correlation coefficient for logaritmic approximation.

Quite large error bars of distance moduli are (Table 6 / pp. 69-70)
from minimum +- 0.38 to +- 1.33 magnitudes.

Anyhow comparing my previous article formulae (data from Supernovae Type Ia,
(0 < z < 1.390)

SNIa data 2006 (0 < z < 1.390):
------------------------------

u = 2.3935*ln(z) + 44.172 ,

R^2 = 0.9914 , correlation coefficient for logaritmic approximation.

SNIa dada 2007 uppdated (0 < z < 1.390):
---------------------------------------

u = 2.4175*ln(z) + 44.163 ,

R^2 = 0.9948 , correlation coefficient for logaritmic approximation.

These GRB measurements from reference 1 (even though quite large error bars)
at least give approximately same approximate functions.

References:

1. Schaefer Bradley, 2006.
The Hubble Diagram to Redshift > 6 from 69 Gamma-Ray Bursts.
arXiv:astro-ph/0612285v1 11 Dec 2006.
Table 6, pp. 69-70, pp. 59-88, 88 pages.

2. Riess Adam G. et al. 2007.
New Hubble Space Telescope Discoveries of type Ia Supernovae at z => 1:
Narrowing constraints in the Early Behaviour of Dark Energy.
The Astrophysical Journal 659:98-121, 2007 April 10.
pages 106-111. Table 4 on page 109 (distance normalization is arbitrary,
0 < z < 1.390 ), Figure 6 on page 110.

3. SN Ia 2006 data / SN Ia 2007 updated data mentioned above article
is taken from the reference:

http://braeburn.pha.jhu.edu/~ariess/R06 .

4. Poropudas Hannu, 2016.
Extiction-corrected Distance Modulus u_o as a function of Redshift z of SN Ia data 2006 and 2007.
sci.physics.relativity,
Thu, 14 Jan 2016 01:16:47 -0800 (PST),
Message-ID: <d08cf609-461c-44a3...@googlegroups.com>.

5. Poropudas Hannu, 2016.
Extinction-corrected Distance Modulus u_0 as a function of Redshift z.
sci.astro.research,
Sat, 16 Jan 2016 19:30:14 -0800 (PST),
Approved: sci.astro.res...@slimy.greenend.org.uk (jt),
Message-ID: <74a3be85-d6e1-4ae6...@googlegroups.com>.
(same article as 4.)



Best Regards,

Hannu Poropudas

Kolamäentie 9E,
90900 Kiiminki / Oulu,
Finland

[Jack...@hotmail.com]

On Mon, 8 Feb 2016 22:58:27 -0800 (PST), Hannu Poropudas
<hanp...@luukku.com> wrote:

>Approximating function to The Hubble Diagram to Redshift > 6 from 69 Gamma-Ray Bursts
>
>Distance moduli
>u as function of redshift z.
>
>u = m - M = 5* log (d_L) + 25,

Needs a +

>(d_L in units of megaparsecs),
>
>d_L = luminosity distance,
>
>u = distance moduli,
>m = apparent magnitude,
>M = absolute magnitude.

You are on the wrong track. You want to stick with log() not ln()
because of how magnitudes work. A 1st magnitude star is 100 times
brighter than a 6th magnitude star, so the ratio between magnitudes is
2.51. 2.5 to the 5th power equals 100. The 5(.02) is in there to
account for distance squared.

The factor 25, when divided by 5 = 5, the log of 100,000. That's to
cover the difference between 10 parsecs and mega parsecs.
As you see the 25 is not an arbitrary coefficient that can be
corrected as you do with your 44.159.

No, there needs to be another legitimate and fundamental term that
attenuates the distance modulus to match the supernovae.(as a start,
because the lifetime stretching anomaly still has to be disposed of).

It cannot be found in the literature but I have derived it Howard that
damn send buttonand it accounts for both anomalies. it is well
concealed in the equations, but it is straightforward.
Cosmological constant and the dark energy are just quizzical
hypotheses based on LCD M and FLRW fictions.
John Polasek













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