cosmological model with variable light velocity

Variable speed of light cosmological model. Comparison with observational data on QSOs.

Jean-Pierre Petit

and Maurice Viton

(Laboratoire d'Astronomie Spatiale. Traverse du Siphon. 13012. Marseille)

Physics Letters A, Vol. 4, n°23 (1989) pp. 2201-2210

ABSTRACT :

After a complement to previous articles on the gauge invariance of the Boltzmann collision operator, we compare a recent homogeneous set of data on radio QSOs, including angular sizes and lobe curvatures, with what is expected from either our new cosmological gauge model or the most commonly accepted Friedmann model with q₀ = 1/2. It is shown that the new gauge model provides a much better fit to the angular size distribution versus redshift and similarly to the curvature, thanks to simplified assumptions on the mechanisms involved in the formation of jets.

1. INTRODUCTION

In references [1] and [2], hereafter paper I and paper II respectively, we have previously developed a cosmological model in which all the so-called physical constants were made free, so that we had to introduce new physical laws, gauge laws, in order to conveniently link these constants:

c (speed of light), G (gravitation), h (Planck's constant), mₑ (electron mass), mₚ, mₙ (proton and neutron mass). It was first shown that general relativity theory does not require the absolute constancy of G and c, but only the absolute constancy of the ratio G/c² (Einstein's constant of the field equation). This gave the first linking relation. The other came from geometric considerations: we assumed that the characteristic lengths such as the Jeans length, the Schwarzschild length and the Compton length followed the variation of the scale parameter R(t).

By combining these new physical laws, we obtained the following relations:

(1)

(2) mₚ = mₙ (nucleon mass)

»

R(3)

h

»

R 3/2

(4)

G

»

1/R

(5)

V (velocity)

»

R⁻¹/²

(6)

r

»

1/R ²

In addition, we found a single cosmological model, with negative curvature, filled indifferently with photons or matter or a mixture of the two, with non-zero pressure, and obeying the single law:

(7a)

In paper II, we rediscovered the Hubble law, due to the secular variation of the Planck constant (which was found to vary like t) and not to the expansion process. In this constant energy model, geometric considerations made some characteristic energies such as the ionization energy vary like R(t), and it was found to be consistent with additional gauge relations applied to electromagnetism. It then appeared possible to derive the distances of light sources from the redshift data. They were found to be quite close, for moderate z values, to the classical values derived from a Friedmann model with q₀ = 1/2, since the ratio:

(7b)

remains close to unity within 5 percent for z ≤ 2.

2. A SHORT COMPLEMENT ON GAUGE INVARIANCE

In paper I, section 5, we showed that some fundamental equations (Vlasov, Schrödinger, Maxwell) were invariant under the proposed gauge relations. Let us show that the Boltzmann collision operator is also invariant.

Write this equation.

(8)

f is the velocity distribution function, g is the relative velocity of two particles during a collision, b is a length (impact parameter), e an angle. Introduce dimensionless variables, through:

(9)

t = t* t ; f = f* x ; V = V* w ; g = V* g ; r = R* z Y = (Gm/R*) j ; b = R* b

The characteristic velocity distribution function is:

(10)

Following the gauge relations as defined in paper I, Gm/R* varies like 1/R*. V* varies like 1/R¹/². m varies like R* (mV² is constant). The energy kT is constant. In summary, f* » R*⁻³/², and therefore:

(11)

Such a dimensional analysis gives terms varying respectively like 1/t*, V*/R* = 1/R³/², 1/R³/², which implies again:

(12)

R* » t*²/³

and hence the invariance of the Boltzmann operator.

3. OBSERVATIONAL TESTS.

After this brief parenthesis, let us turn to a comparison between various models and the radio data on 134 QSOs recently published by Barthel and Miley [3, hereafter BM], in which they show that distant DSOs have smaller angular sizes, larger curvatures and higher luminosities than those nearby. Note, however, that we do not intend to discuss the given intrinsic powers here, since the physical mechanisms involved in the generation of relativistic jets are not yet clearly understood.

The situation seems apparently simpler regarding the angular size and curvature of radio sources, since geometric properties are mainly at play in both cases, a priori, although we cannot ignore that important systematic effects might be at work, and we refer the reader to the detailed discussion of BM on the mechanisms involved. In short:

• interaction with the intergalactic medium (IGM) can efficiently disrupt initially collimated jets, leading to the formation of large, turbulent lobes of lesser extent: if it is clear that such effects can significantly modify the angular size distribution at a given redshift, more complex mechanisms have been invoked by BM to explain the stronger curvature of lobes observed at large redshifts.

• possible evolutionary effects in all the elementary mechanisms mentioned above, including gauge processes not yet identified.

• an observational bias such as the well-known Malmquist bias, introducing an underestimation of angular sizes for distant QSOs.

Now, let us suppose in this article that these potential effects are not dominant in the data, i.e. that the distribution of angular size and curvature versus redshift can be considered as a good test to discriminate between different cosmological models, and show that the new gauge model provides a better fit to these distributions than the classical models.

3a The angular size.

The angular size of extragalactic objects has often been considered as a powerful test for cosmological models. Let the subscript 1 be associated with the emission epoch and the subscript 2 with the reception epoch: since the light emitted by the edges of a source at time t₁ follows radial trajectories, the angular size φ is conserved for a current observer, so that we can write classically, regardless of the model:

(13)

where D(t₁) is the linear diameter of the source and d(t₁) its metric distance.

In the classical Friedmann model with q₀ = 1/2 (the famous Einstein-de-Sitter model), D(t₁) = Dconstant, R(t₁) = R(t₂)/(1+z) and d(t₁) = R(t₁) u where:

(14)

and therefore a certain paradox appears, since the angular size obeys:

(15)

This function has a minimum for z = 1.25, then tends to increase linearly with z.

Now, with the new gauge model, we have D(t₁) = D(t₂)/(1+z), R(t₁) = R(t₂)/(1+z) and d(t₁) = R(t₁) u as well, but with:

(16)

and the angular size obeys:

(17)

When z tends to infinity, φ tends to a constant, a clearly different behavior and in qualitative agreement with the data.

(18)

Fig 18: The largest angular size (LAS, in arcseconds), on a logarithmic scale versus redshift for the 95 extended sources with "T", "D1" and "D2" morphologies (squares) and the 33 compact sources with "SSC" morphology (asterisks). The two curves represent the fits of the gauge model (continuous line) and the Einstein-de-Sitter model (dotted line) derived for extended sources in this paper. Three additional radio sources of very large extent are shown for comparison. 4C41.17 being the farthest galaxy currently known, 4C74.26 the largest radio source associated with a quasar.

In order to quantitatively compare the quality of fit of the two models to the data, write f = f₀ f(z), where f(z) is a characteristic function predicted above by each model. Linear regressions have been performed between f(z) and the "largest angular size" [LAS after BM] data, either for the complete sample of 134 QSOs, or for a reduced sample of 83 QSOs in which only two-sided lobe sources were selected, i.e. those with the "T" or "D₁" morphologies defined by BM, excluding thus the steep spectrum core or "SSC" and the one-sided lobe or "D₂" sources, in order to test possibly different properties between compact and extended sources. The results of the regressions are as follows, all the given linear coefficients and their RMS error bars being expressed in arcseconds:

• with the Einstein-de-Sitter model:

F = (28.8 ± 2.9) f(z) - (89.3 ± 20.1)

for the complete sample, and

F = (31.3 ± 3.3) f(z) - (90.5 ± 20.8)

for the reduced sample.

• with the gauge model:

F = (21.5 ± 2.11) f(z) - (19.1 ± 19.6)

for the complete sample, and:

F = (24.3 ± 2.3) f(z) - (17.3 ± 19.8)

for the reduced sample.

It is clear that the new gauge model provides a much better fit to the data, since regardless of the sample, the moderate constant term it implies is marginally significant from a statistical point of view, and therefore the (expected) zero value is very likely. The situation is completely opposite with the classical model, here as well regardless of the sample, since the constant term is highly significant from a statistical point of view and its large, negative value is unacceptable on theoretical grounds, unless one assumes that very strong systematic effects, as those suspected above, are at work in the data.

3b. The curvature.

Let us also show that the more curved, distorted appearance of distant QSOs highlighted by BM can be strangely explained by the new gauge model, provided it is not an artifact resulting from various systematic effects. Since in the new model we assume that all the energies are conserved during the cosmic gauge process, we can include the conservation of the rotational energy of the QSO core emitting the jets:

(19)

As m » R, I » R³ and W » R⁻³/², W » 1/t » (1+z)³/²,

in curious agreement with the one-dimensional lower law fitted by BM on the reduced sample (since the curvature is only defined in this latter case), that is:

(20)

Fig 21: The curvature (in degrees) on a logarithmic scale versus redshift for the 83 extended sources with "T" and "D1" morphologies only, for which it is defined. The continuous curve corresponds to the fit of the gauge model, indicating that angular speeds were higher in the past, while the dotted line represents the power law fit of Barthel and Miley (BM)

.

Now, let us propose a rudimentary explanation, referring to the recent analysis of Greyber [5] on the nature of the central engines in QSOs responsible for their enormous energy production rates: if we accept the figure (i) that the plasma blobs are ejected at high velocities from the central engine, continuously or not, along the magnetic dipole axis of the QSOs and (ii) that this latter is not generally coincident with the angular momentum axis, we are faced with a model similar to a rotating "garden sprinkler", in which the jets will bend into some kind of an Archimedean spiral, as long as the interaction with the IGM remains negligible. And even if this interaction becomes significant at some distance from the nucleus, the jets will stop expanding there, resulting in an increased curvature. Since there is no reason why the IGM density would be spherically distributed around QSOs, such interactions could explain the frequently asymmetrical jets as well as random effects on the overall curvature, as discussed by BM.

Therefore, the higher the redshift of the QSO, the higher its angular velocity due to the cosmic gauge process, and hence the greater the curvature of its jets.

4. CONCLUSION.

We have focused on specific features recently highlighted in the distribution of angular sizes and curvature versus redshift for a homogeneous set of 134 radio QSOs. We found interestingly that our gauge model with "variable constants" provides a better fit to these distributions than the conventional Einstein-de-Sitter model, provided (i) that the observed trends (smaller angular size and larger curvature of distant QSOs) are confirmed by future observations, (ii) that these trends are not dominated by various effects or artifacts, and (iii) that the rudimentary assumptions made on some of the mechanisms involved are real. Furthermore, further investigations on the intrinsic power of these sources are necessary to understand if the gauge model provides a better understanding of the observed trends, i.e. that the luminosity of distant QSOs is indeed much greater than that of nearby QSOs.

5. REFERENCES.

[1] J.P. PETIT : Interpretation of cosmological model with variable light velocity. Modern Physics Letters A, Vol. 3, n° 16, November 1988

[2] J.P. PETIT : Cosmological model with variable light velocity. The interpretation of red shifts. Modern Physics Letters A, Vol. 3, n° 18, December 1988.

[3] P.D. BARTHEL & G. K. MILEY. Evolution of the radio structure in quasars: a new probe of protogalaxies? - Nature Vol 333, 26 May 1988.

[4] M.L. NORMAN, J.O. BURNS and M.E. SULKANEN : Disruption of galactic radio jets by shocks in the ambient medium, Nature 335 (1988) 146.

[5] H.D. GREYBER : The importance of strong magnetic fields in the Universe, Comments Astrophys. 13 (1989) 201.