 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 made by the Standard Model of Cosmology (ΛCDM) based on measurements of the cosmic microwave background radiation.  The Hubble 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 Tired-Light (STL) model with the Spectral Transfer Redshift (STz).  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.

The calculation includes three massless neutrino species and a correction for annihilations of particles not present now (e.g. e+/e ).

km/s/Mpc

Open: uses ΩΛ = 0 giving an open Universe [if you entered ΩM < 1].
Flat:     uses 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' scaling relations.

## ΛCDM Cosmology

*** JavaScript not enabled or script file missing ***

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

## STL Cosmology

An equivalent to the Mattig formula is obtained in a Static Universe Cosmology from the differential equation  -d(hν) = H(hν) dD/c  expressing
the energy loss as a function of distance for a Tired-Light redshift, with the solution:

DA = (c/H0) ln (1 + z).

Since the universe is flat and static, 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+y)/2 DA.

For the transient signal of a Type Ia supernova, STz Tired-Light modulates the observed spectrum with the rest-frame light-curve of duration ∝ λy.

An extended object at temperature T is seen as a blackbody at the Wien temperature TW(z) = T/(1 + z), but with a spectral radiance increased by a factor of (1 + z)3

B00, TW)  =  (1 + z)3  2hc2 / [λ05 (ehc/(λ0 k TW) - 1)].

/Gpc

Flat Universe

## STL Cosmology

*** JavaScript not enabled or script file missing ***

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

## 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 one concludes: "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 their stars appear to have a blackbody spectrum with an excess brightness.

Updated 2021-10-20  