Accelerating Expansion of the Universe: Yes or No?
| Scientific Paper | |
|---|---|
| Title | Accelerating Expansion of the Universe: Yes or No? |
| Read in full | Link to paper |
| Author(s) | Bob Day |
| Keywords | Expanding Universe, Acceleration, Dynamic Universe, Cosmological constant |
| Published | 2007 |
| No. of pages | 12 |
Read the full paper here
Abstract
Over the past decade most cosmologists have come to believe that the universe is expanding at an accelerating rate. Seeming evidence for this acceleration has been provided by brightness measurements of many newly discovered type Ia Supernovae. These supernovae were discovered to be unexpectedly faint for the distance at which they were according to standard cosmology. If their distances were indeed correct, the unexpected faintness of the supernovae would indicate that the universe is larger than thought. This could be explained by postulating an accelerating expansion, which is the way standard cosmology explains it. Another theory, the Dynamic Universe (DU) theory explains the same measurements just as well as standard cosmology, but without the need for the expansion of the universe to be accelerating. In this essay I will describe both approaches.
Overview
This is an expository essay rather than a research paper. Bob Day sets two cosmological models side by side against the same Type Ia supernova data — the concordance model of standard cosmology, and Tuomo Suntola's Dynamic Universe (DU) — and asks which does better. The theory is Suntola's; Day's role is to present it clearly and to run the comparison, and he does so at a deliberately accessible level, defining magnitude, flux, redshift and distance modulus as he goes and taking care to distinguish the equations that are matters of definition ("there is no controversy about these equations") from those that carry theoretical commitments.
The result Day reaches is a modest one, and he states it modestly. Both models fit the data almost perfectly, with goodness-of-fit numbers of 31.5859 for standard cosmology and 31.5262 for the DU — "a smidge better (smaller) for the DU, but the difference is so small that the normal random inaccuracy in the measurement of a single data point could probably account for it." The claimed advantage is therefore not in the fit but in the accounting: standard cosmology needs two adjustable parameters and a postulated repulsive substance, while the DU reproduces the same curve with one adjustable parameter and no dark energy at all. The departure from the mainstream account is thus not that the universe is not expanding — the DU is an expanding model — but that its expansion need not be accelerating.
The argument
The data and the standard curve
Day uses the 115 Type Ia supernovae published in 2005 by Pierre Astier's group, plotted as distance modulus μ = m − M against redshift z. Standard cosmology relates the two through μ = 5 log10dL + 25, with the luminosity distance given by the familiar integral over 1/√[(1+z)2(1+Ωmz) − z(2+z)Ωλ], citing Carroll, Press and Turner (1992). With c and H0 = 70 km/s/Mpc fixed and flatness imposing Ωm + Ωλ = 1, the best fit is Ωm = 0.280, Ωλ = 0.720. Since acceleration requires Ωλ > ½Ωm, and 0.72 > 0.14, the fit implies acceleration — "IF you believe the theory, that is." Day's complaint at this point is epistemic rather than technical: the acceleration is inferred through the theory rather than measured, "we have no way of independently checking it", and dark energy is "just a postulate".
The Dynamic Universe distance
The DU treats space as the three-dimensional surface of an expanding four-sphere of radius R, expanding along the fourth dimension at exactly the speed of light. Because the velocity of light in space equals the velocity of space in the imaginary direction, the "optical distance" dO that light covers equals the growth of the 4-radius during the journey: dO = R − R1. Wavelength stretches with the expansion, so z = (R − R1)/R1, giving dO = zR1 and 1 + z = R/R1, and hence
- dO = zR/(1 + z)
with R treatable as a constant over human timescales.
The DU magnitude equation
The DU computes magnitude from observed flux directly, m = −2.5 log10(Fobs/Fr) with Fr = 2.53×10−8 W/m2, and builds Fobs from four factors: the supernova luminosity Le ≈ 1.595×1036 W; inverse-square dilution 1/(4πdO2); a single factor 1/(1+z) for the increased spacing of the photons; and a further 1/(1+z)2 inserted "to remove the K-correction" that observers have already applied to the published data. Notably, in the DU the stretching of a photon's wavelength "does not alter their energy, it merely spreads it out over a longer distance (if that were not the case, their energy would not be conserved!)" — hence only one power of (1+z), not two, from the expansion itself.
Combining these, the z-dependence collapses to Fobs ∝ 1/[z2(1+z)], and the magnitude becomes
- μ = M + 5 log10(z) + 2.5 log10(1+z) + C
with a single fitted additive constant, C = 43.330. The shape of the curve is fixed by the theory; only its vertical placement is free.
The comparison
Overlaid, the two curves are "all but identical". Day's argument then turns on parsimony: standard cosmology adjusts both a vertical offset and a shape parameter (Ωλ), the DU only the offset. He explains the point with a clean analogy — a parabola fitted to points scattered about a straight line will always beat the line on goodness of fit, because the scatter will supply "just a tad of curvature" — and concludes that the DU's equal fit with fewer knobs is the stronger result. He closes by listing effects the DU is said to reproduce alongside general relativity: mass–energy equivalence, the perihelion precession of Mercury, light bending near stars, and gravitational clock slowing.
Assessment
The essay is well made on its own terms, and its arithmetic holds up under checking. The DU flux chain reduces correctly: Fobs ∝ (1+z)2/z2 × (1+z)−1 × (1+z)−2 = 1/[z2(1+z)], which on taking −2.5 log10 gives exactly the 5 log z + 2.5 log(1+z) that Day quotes. His constants check out too: an absolute magnitude of −19.31 corresponds to 3.83×1026 × 10(4.74+19.31)/2.5 = 1.596×1036 W, matching his 1.595×1036, and his reference flux of 2.53×10−8 W/m2 is the standard bolometric zero point. The condition Ωλ > ½Ωm for present-day acceleration is correct, as is the flat-universe constraint. The parameter count is also fair as stated: with H0 fixed and R absorbed into C, standard cosmology fits one offset plus one shape parameter, the DU one offset only. Day does not overclaim from the tiny goodness-of-fit difference, and he says outright that it is within the noise. That restraint is worth noting, because the same comparison is routinely oversold elsewhere.
The most serious weakness is a step Day flags himself but does not follow through. The factor 1/(1+z)2 "to remove the K-correction" is described as "actually not precise, but a very good approximation", and it is entirely load-bearing. Drop it and the DU curve becomes μ = 5 log z − 2.5 log(1+z), which at z = 1 differs from the fitted form by 1.5 magnitudes — far more than the scatter in the data. A correction that flips the sign of the only shape term in the model cannot be waved through as an approximation, and it is applied to one model and not the other, even though both are being fitted to the same published, already-K-corrected distance moduli. The real K-correction depends on the observed and rest-frame passbands and on the supernova's spectral energy distribution, and is not a pure power of (1+z) at all. Until that step is done properly, the claim that the DU's shape is "fixed by the theory" rather than adjusted is not established.
The second difficulty is one of scope. Day writes that for accelerating expansion "we have no way of independently checking it", and this was already not the case in 2007. The value Ωλ ≈ 0.7 is not extracted from supernovae alone: it follows independently from the angular positions and relative heights of the acoustic peaks in the cosmic microwave background measured by WMAP, and from the baryon acoustic oscillation scale detected in the SDSS and 2dF galaxy redshift surveys in 2005 — three datasets of quite different physics converging on the same value. A model that matches the supernova Hubble diagram has therefore cleared only one of three hurdles, and the essay does not attempt the other two. The same applies to the closing list of general-relativistic successes: reproducing Mercury's perihelion and light bending is necessary but does not touch the cosmological question at issue.
Third, a point on the parsimony argument. It is a good argument, and Day states it well, but the accounting is incomplete on both sides. The DU's single free constant comes only after fixing the four-sphere geometry, the expansion rate at exactly c, and the photon-energy convention that supplies one power of (1+z) instead of two — postulates that are no more independently checked than dark energy is. Conversely, Ωm in the standard model is not a free knob in practice; it is constrained by galaxy cluster masses and by the CMB. Counting fitted coefficients in a single plot is a weaker test than it appears when both models bring unfitted structural assumptions to the comparison.
Finally, the essay is candid about what it is: a comparison on one dataset, presented for a general reader, with the theory attributed to Suntola and the figures acknowledged. It does not claim the DU is proved, only that on this evidence it holds a "small advantage". That is a defensible thing to say about the supernova Hubble diagram taken in isolation. It is not a defensible thing to say about the concordance model as a whole, and the essay would be stronger if it said which of the two it meant.