 Original image by NASA, ESA, H. Teplitz and M. Rafelski (IPAC/Caltech), A. Koekemoer (STScI), R. Windhorst (Arizona State University), and Z. Levay (STScI); cropped by L. Marmet.

# Cosmology Calculator

Direct measurements of the Hubble constant disagree with predictions from ΛCDM based on measurements of the cosmic microwave background radiation.  The tension is forcing cosmologists to question every aspect of the ΛCDM model.

This webpage provides two Cosmology Calculators: one for the ΛCDM model, the other for a Static Universe cosmology in which a Tired-Light mechanism produces the cosmological redshift.  Enter values and the results will appear immediately.

## ΛCDM Cosmology

Based on the calculator written by Ned Wright (2006, PASP, 118, 1711):
http://www.astro.ucla.edu/~wright/CosmoCalc.html.

The size evolution of star-forming galaxies follows the power law:

re = r0 (1 + z)nS kpc.

The Tolman test for expansion predicts a surface brightness <SB> that decreases as (1 + z)4. The luminosity evolution of galaxies produces the Tolman signal

Δ<SB> = 2.5 log (1 + z)4-nL mag,
ΔMevol = 2.5 log (1 +z)nL mag.

These two functions are included below so that the calculated distances correspond to raw observational data.

km/s/Mpc

Open:  sets ΩΛ = 0 giving an open Universe [if you entered ΩM < 1].
Flat:     sets ΩΛ = 1 - ΩM giving a flat Universe.
General: uses the ΩΛ that you entered.

used for sample selection
introduces a bias, c.f. 'Fundamental Plane' for elliptical galaxies.

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Source for the default parameters H0, ΩM and ΩΛ
Source for the default parameter nS
Source for the default parameter nL

## SUTL Cosmology

An equivalent to the Mattig formula is obtained in a Static Universe Cosmology from the differential equation -d(hν) = H(hν) dt expressing the energy loss as a function of time for a Tired-Light redshift.
Defining D as the radial distance gives dD = -c dt and the solution:

DA = (c/H0) ln (1 + z).

Since the universe is flat and not expanding in the model, the radial distance is equal to the angular distance DA.  The energy lost as (1 + z)-1/2 by the redshift mechanism results in a bolometric luminosity distance:

DL = (1 + z)1/2 DA.

An object at temperature T is seen as a blackbody with excess brightness

E00)  =  (1 + z)3 8πhc / [λ05 (ehc/(λ0kTW) - 1)]

but the Wien temperature TW(z) = T/(1 + z) equals that predicted by ΛCDM.

/Gpc

Flat Universe

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Source for the default parameter Hz ≡ H0/c
all parameters to default values.
the value of H0 that gives raw dA(ΛCDM) = dA(SUTL).

## Discussion

It is difficult to compare the two models to a high accuracy. Astrophysical data are currently analyzed with the assumptions of ΛCDM cosmology and corrected for various effects in powers of 1 + z (e.g. luminous flux, time dilation, galactic evolution, Hubble residuals, deceleration parameter, etc.)

Both calculators give observed quantities which, for ΛCDM, are corrected to include the large effects of galactic evolution. However other effects are difficult to disentangle from an analysis based on the ΛCDM model.

Interpreting observations with the ΛCDM model, it is easy to understand why, "compared to typical galaxies at later times, [young galaxies in the first billion years of cosmic time] show more extreme emission-line properties, higher star formation rates, lower masses, and smaller sizes." [Ref.]  In a static universe, galaxies are farther away and appear to have a blackbody spectrum with an excess brightness.

Updated 2020-10-5  