Supernova Redshift, et al

Charles Francis

As discussed on s.a.r. Bob Day has used Matlab to test the distance red-
shift relation found by replacing the affine connection of gtr with a
teleconnection. The results are posted on the bottom of his home page at
http://bobday.vze.com. They are:

For the first year data from The Supernova Legacy Survey, Astier
arXiv:astro-ph/0510447 he found:

Standard model optimum Omega_m = 0.263, chi-square = 113.5679
teleconnection model, optimum Omega_m = 1.06, chi-square = 115.2288

I believe chi^2=115 on a sample of 115 is 1sd, the expected result if
error margins are correctly stated, showing that the teleconnection
model fully accounts for missing mass.

This is the text of my paper, excluding the graphs and the mathematical
appendix. I have used W instead of Omega, throughout.

Does a Teleconnection between Quantum States account for Missing Mass,
Galaxy Ageing, Supernova Redshift, MOND, and Pioneer Blue-shift?


Abstract:
There have been previous suggestions, notably by Einstein, that the
affine connection in general relativity might be replaced with a
teleparallel one. This paper carries out a preliminary investigation of
the empirical implications of a teleparallel displacement of momentum
between initial and final quantum states, using conformally flat quantum
coordinates. An exact formulation is possible in an FRW cosmology in
which cosmological redshift is given by 1+z = (a_o/a(t))^2.. This is
consistent with observation for a universe expanding at half the rate
and twice as old as indicated by a linear law, and, in consequence,
requiring a quarter of the critical density for closure. Supernova
redshifts indicate a universe a little over critical density and are
consistent with zero cosmological constant. Quantum coordinates exhibit
an acceleration in time, resulting in the anomalous Pioneer blue-shift
and in the flattening of galaxies' rotation curves. These appear as
optical effects and do not affect classical motions. Milgrom's
phenomenological law (MOND) is precisely obeyed.


1 Background
There is growing concern in Cosmology about unexplained empirical
phenomena. The standard model of accelerating expansion is successful in
matching parameters to observation, but, while there is no true
reconciliation between general relativity and quantum mechanics, science
should remain open to the prospect that these phenomena may have some
deep underlying reason in new physics. Any new model of physics should
adhere to fundamental principles such as the cosmological principle and
the principle of relativity, but the true test is whether predictions
match observation, and whether a model is capable of making new
predictions or providing explanations where previously there were none.
This paper carries out a preliminary investigation of the empirical
implications of a modification to general relativity adhering to
fundamental principles. The tests described here could have falsified
the model, but an initial analysis of current astronomical data is
consistent with a universe of just above critical mass, with no cold
dark matter or cosmological constant, and no apparent timescale problem.
As a bonus it models the flattening of galaxies' rotation curves and
Pioneer blue-shift, without requiring either a change to Newtonian
dynamics or galactic haloes (table 1).

Table 1: Properties Compared
Connection Affine Teleconnection
Topology Open Any
W 0.26+-0.05 1.06+-0.15
WL 0.74-+0.05 -0.1-+0.2
WB 0.025-0.05 0.1-0.2
Age of universe 14X10^9 yrs 16-20X10^9yrs
Age at z=6 7x10^8yrs 4.5x10^9yrs
Baryon:Non-baryon ~15:1 4-9:1
Expansion rate adot/a=H0 adot/a=H0/2
Pioneer blue shift unexplained ap=H0c
MOND CDM aM=H0c/8
wave motion curved space? flat space
classical motion geodesic geodesic

Table 2: Magnitude-Redshift Relation
first order comparison
affine: m-M ~ 5logz + 1.086(1-q)z
teleconnection: m-M ~ 5logz + 1.086(2-q)z

It is known on theoretical grounds that new physics is required to
reconcile general relativity and quantum theory (Dirac, 1964). Eppley
and Hannah (1977) showed that if gravitational measurement causes wave
function collapse in curved space, violation of the uncertainty
relationships can only be avoided by giving up conservation of momentum.
Individual detection of photons from distant stars strongly suggests
that we cannot be certain of the interpretation of redshift without
first having a rigorous formulation of quantum motions in curved space
time.

Einstein (1930) found problems with electrodynamics in curved space
time, and suggested that the affine connection used in general
relativity might be replaced with a teleparallel connection. Such a
replacement can be motivated in the orthodox interpretation of quantum
mechanics; if it does not make sense to talk of position between
measurements then it is also without sense to talk of geodesic motion of
a photon emitted from a distant star and detected on Earth. Since the
connection is meaningful only at the times of measurement it will be
called a teleconnection. Standard general relativity and quantum
mechanics are assumed, excepting that wave functions are defined using
quantum coordinates (section B3), not in curved spacetime.

A heuristic description of a teleconnection is given in section 2. This
should be sufficient to understand the empirical tests described in
sections 3 to 7. A formal treatment is deferred to Appendix B,
Mathematical Description of the Teleconnection, showing that it can be
consistently defined in an FRW cosmology, that the prescription reduces
to the standard affine connection in the classical correspondence, and
that geodesic motion obtains for classical particles and for a beam of
light). Graphical display of comparison of data with theory is given
in Appendix A, Hubble Diagrams for Type Ia Supernovae.

A closed universe with zero cosmological constant and no cold dark
matter will be discussed. Other models are possible but up to the
accuracy of the tests applied here this simple model is consistent with
data, gives accounts of observed phenomena which the standard model has
been unable to explain and makes clear predictions about future tests.
In the instance of galactic rotation curves testing has been done, not
through direct statistical analysis, but by deriving a general law
(MOND) already established from statistical analysis. The distance-
redshift relation has been analysed by Bob Day using data sets from
Riess (2004) and Astier (2005) and a best fit has been found with a
universe with just over critical mass.


2 The Teleconnection
This is _not_ a teleparallel theory using the Weitzenbrock connection
(see e.g. Arcos and Pereira, 2004). Torsion will be removed as part of
wave function collapse and in the classical correspondence gravity will
be described by curvature, as is normal in general relativity. In
general relativity it is assumed that photon momentum is parallel
transported through large distances. This assumption takes no account of
the propagation of a photon wave function in a curved space-time, which
would imply that a photon of precise momentum at time of emission would
not have precise momentum at absorption. Here it is assumed that there
exists a coordinate space in which plane wave states are defined (B3.1).
Momentum at source is teleparallel to momentum at detection, and this
determines a connection between the initial and final states.

In classical general relativity there is no local meaning to expansion
because length is defined locally, by an empirical procedure based on
local matter. To talk about expansion we have to compare a length scale
defined here and now, using here and now clocks and rulers, with a
length scale defined at some time in the past. In practice we can do
this by studying light from the past and analysing redshift provided
that we know how light behaves. The definition of a teleconnection
assumes that if momentum has a precise value at one place and time then
it also has a precise value other places and times and is empirically
justified in so far as observation yields precise values for
cosmological redshift after allowing for dispersion due to dust or other
known factors. This is a fundamental assumption in this model, of equal
importance to the assumption of the constancy of the speed of light in
special relativity. Like that assumption, if it were dropped we would be
left, not with a different theory, but with no known consistent theory.

Let Alf be an observer on a space craft or a distant planet, and let
Beth be an observer on Earth, such that Alf can signal to Beth. At the
time of emission of a photon passing from Alf to Beth, Alf defines
synchronous, conformally flat, co-ordinates in 3 dimensions at constant
cosmic time t. In a closed cosmos the universe can mapped onto a finite
space, which will be called Alf's map. Beth defines Beth's map in
exactly the same way, to the same scale, at the time of detection of the
photon, cosmic time t0. For a closed universe in three dimensions Alf's
and Beth's maps each consist of the interior of a sphere. Let a(t) be
the scale factor and let a0=a(t). If the universe expands during the
time of travel of the photon from Alf to Beth, then Beth's map is larger
than Alf's map. Because the maps are conformally flat, they can be
placed in direct correspondence by enlarging Alf's map by a factor
a0/a(t). The teleconnection is defined such that photon momentum is
represented by an arrow of equal length and direction on Beth's map and
on Alf's enlarged map.

Quantum coordinates define a four dimensional map found by considering
all the times and positions where Alf and Beth might be, with the time
axis scaled so that light is shown at 450. This is a Penrose diagram in
each time-radial plane. In these coordinates the arrow representing
photon momentum is of constant length and direction everywhere, so that
plane wave motions obtain for light. Beth can compare the scale of her
map to that of Alf's map by studying red shift. There are two scaling
effects. First Alf's map has been enlarged by a factor a0/a(t). In
addition, the scaling on the map changes as you move from one point to
another. That gives another factor a0/a(t). Thus, the model predicts
that the cosmological redshift factor varies with the square of the
expansion parameter.

1+z=a0^2/a^2(t) 2.1

On Beth's map, Alf, and all physical objects in Alf's locality such as
rulers, appear enlarged. This is torsion. In measurements in quantum
mechanics there is both an initial and a final measurement and the
coordinate system is scaled to the measuring apparatus at the time of
each measurement. Rescaling coordinates removes torsion and renormalises
momentum so that in the classical correspondence gravitational redshift
is as in general relativity, as required by the principle of equivalence
and for geodesic motion (appendix B4).

3 Cosmological Redshift
Typically in quantum theory experiments require a measurement of the
initial state and a measurement for the final state, and are such that
reference matter used for the initial measurement is rigidly related to
that used for the final one; either the same coordinate axes and clock
are used in both measurements or the coordinate axes and clocks are
calibrated to each other. As seen in section B4 this requires a
renormalisation of energy momentum, such that geodesic motion and the
principle of equivalence are restored. But in measurements on light from
a distant object it is not possible to define a prior relationship
between the reference matter used for the final measurement and the
matter from which the photon is emitted. Light received at the origin
has been transmitted from an event on the light cone, so that the only
information we have about the initial state comes from measurement of
the final state. There is then no renormalisation of energy-momentum,
and cosmological redshift is given by 2.1. For small r

1+z = 1 + 2r adot/a 3.1

Thus coordinates in which radial distance from Earth is calculated from
redshift exhibit a stretch of factor two in the radial direction. The
time taken for a pulse of light to traverse a small angular distance
dtheta is rdtheta, so that there is a stretch of factor half in the
angular direction (this gives 4pi in a circle, which may cast light on
fermion phase under rotation). Thus the coordinate metric, B3.2, in
quantum coordinates is:

ds^2 = a^2((dt^2 -dro^2)/4 - 4f(ro)(dtheta^2 + sin^2theta dphi^2).
3.2

[without the 4's this should be familiar as the metric in Penrose
coords]

Definition: Hubble's constant, H=2adot/a, is read from 3.1.

It follows immediately that the rate of expansion of the universe is
half that predicted by the standard model, the universe is twice as old
as would be indicated by a linear law, and critical density for closure
is a quarter of the standard value. There is no timescale problem for a
closed universe with greater than critical density and zero cosmological
constant. If observations at high red shift had revealed the expected
activity of the early universe it would have falsified the square red
shift law; in fact it receives support from the observation of mature
galaxies at z=1.4 and greater (e.g. Mullis et al., 2005; Doherty et al
2005, and references cited therein). As described by Glazebrook (2004),
there is poor agreement between current theoretical models of galaxy
evolution and empirical data. To explain this it has been suggested
(Cimatti et. al, 2004) that the theoretical models may be inaccurate.
This model presents an alternative, that a square redshift law means we
have to revise the ages of red galaxies. A value of Hubble's constant
h=0.72 places an upper bound on the age of the universe of eighteen
billion years, so that at redshift 6 the universe would have been about
4.5 billion years old. A detailed study is required to assess
consistency between observation and theory, but this certainly appears
to alleviate the difficulties. Hopefully future observation and analysis
will be conclusive.

4 Cosmological Parameters
The interpretation of redshift alters cosmological parameters but
otherwise leave the classical equations of general relativity unchanged.
Friedmann's equation is

[as usual]. 4.1

Normalising so that W=1 is critical density we define

W=..., W_R=... and W_L=..., 4.2

where k=-1,0,1 (quadruple the standard values, so that W=1 is critical
density). Then Friedmann's equation, 4.1 is,

adot/a = H0/2 (W(1+z)^3/2 + W_R(1+z) + W_L)^1/2 4.3

[standard form has redshift factors squared and no /2], requiring that W
+ W_R + W_L =1. [as standard]

Definition: Let the deceleration parameter be q=-4a adotdot/aH^2

Differentiate Friedmann's equation to find the acceleration equation,

[similarly similar to standard]. 4.4

Let H0=H(t0), a=a(t0) and q0=q(t0). From 4.2 and 4.4,

q=W/2 -W_L 4.5

[standard]

Charles Francis

Thus spake Charles Francis <charles@CF.wanadoo.co.uk>
>
>
>4 Cosmological Parameters
>The interpretation of redshift alters cosmological parameters but
>otherwise leave the classical equations of general relativity unchanged.
>Friedmann's equation is
>
> [as usual]. 4.1
>
>Normalising so that W=1 is critical density we define
>
> W=..., W_R=... and W_L=..., 4.2
>
>where k=-1,0,1 (quadruple the standard values, so that W=1 is critical
>density). Then Friedmann's equation, 4.1 is,
>
> adot/a = H0/2 (W(1+z)^3/2 + W_R(1+z) + W_L)^1/2 4.3
>
>[standard form has redshift factors squared and no /2], requiring that W
>+ W_R + W_L =1. [as standard]
>
>Definition: Let the deceleration parameter be q=-4a adotdot/aH^2
>
>Differentiate Friedmann's equation to find the acceleration equation,
>
> [similarly similar to standard]. 4.4
>
>Let H0=H(t0), a=a(t0) and q0=q(t0). From 4.2 and 4.4,
>
> q=W/2 -W_L 4.5
>
>[standard]
>
[continued]
Angular size distance is calculated as in the standard theory, but
rescaling of coordinates introduces a factor of a0/a(t)=sqrt(1+z), which
adds 2.5log(1+z) to magnitude. To first order in z the magnitude-
redshift relation (c.f. Misner Thorne Wheeler 1973, eq. 39.35b) becomes

m-M ~ 5logz + 1.086(2-q)z 4.6

Charles Francis

Thus spake Charles Francis <charles@CF.wanadoo.co.uk>
>For the first year data from The Supernova Legacy Survey, Astier
>arXiv:astro-ph/0510447 he found:
>
>Standard model optimum Omega_m = 0.263, chi-square = 113.5679

(this is identical to the value of Omega_m given by Astier, confirming
that Bob's program is now correct)

>teleconnection model, optimum Omega_m = 1.06, chi-square = 115.2288
>
I should say, this was for a flat space Lambda model. He has now also
given the optimum value for a Lambda=0 model. Omega_m=1.11, with chi-
square=114.89.

In the teleconnection model I don't think the WMAP analysis can be used
to assert flat space, because are wavefunctions are defined in
conformally flat quantum coords, so this seems to me to be the most
interesting model.


Regards

--
Charles Francis
Please reply by name

Charles Francis

[Moderator's note: Charles, please contact me via email. -P.H.]


Thus spake Charles Francis <charles@CF.wanadoo.co.uk>
>Thus spake Charles Francis <charles@CF.wanadoo.co.uk>
>>
>>
>>4 Cosmological Parameters
>>The interpretation of redshift alters cosmological parameters but
>>otherwise leave the classical equations of general relativity unchanged.
>>Friedmann's equation is
>>
>> [as usual]. 4.1
>>
>>Normalising so that Wppppp1 is critical density we define
>>
>> Wppppp..., W_Rppppp... and W_Lppppp..., 4.2
>>
>>where kppppp-1,0,1 (quadruple the standard values, so that Wppppp1 is critical
>>density). Then Friedmann's equation, 4.1 is,
>>
>> adot/a ppppp H0/2 (W(1+z)^3/2 + W_R(1+z) + W_L)^1/2 4.3
>>
>>[standard form has redshift factors squared and no /2], requiring that W
>>+ W_R + W_L ppppp1. [as standard]
>>
>>Definition: Let the deceleration parameter be qppppp-4a adotdot/aH^2
>>
>>Differentiate Friedmann's equation to find the acceleration equation,
>>
>> [similarly similar to standard]. 4.4
>>
>>Let H0pppppH(t0), apppppa(t0) and q0pppppq(t0). From 4.2 and 4.4,
>>
>> qpppppW/2 -W_L 4.5
>>
>>[standard]
>>
>[continued]
>Angular size distance is calculated as in the standard theory, but
>rescaling of coordinates introduces a factor of a0/a(t)pppppsqrt(1+z), which
>adds 2.5log(1+z) to magnitude. To first order in z the magnitude-
>redshift relation (c.f. Misner Thorne Wheeler 1973, eq. 39.35b) becomes
>
> m-M ~ 5logz + 1.086(2-q)z 4.6
>
[I don't know why this has lost most of the post, but it's the second
time it's happened, so here's hoping it will all come through this time]

Then if qs is the deceleration parameter for the standard model q0~qs+1.
For a flat space model, WpppppWs+2/3. If Lppppp0, Wppppp3Ws. In this case an
estimate of the error may be made by combining the systematic and
statistical errors given by Astier, and multiplying by 3. For critical
density, Wppppp1, WLsppppp0, 4.6 is identical to first order, to the magnitude-
redshift relation with Wsppppp0.33, WLsppppp0.67. The plot for this model has
been superimposed in green on residual Hubble diagrams by adding
2.5log(1+z) to the standard plot (figures 1, 2, 3). These show that, for
Z>0.4 the fit is closer to Wsppppp0.26, WLsppppp0.74, the best fit for the
Supernova Legacy Survey first year data (Astier et al, 2005, figure 1),
than it is to the Wsppppp0.35, WLsppppp0.65 model given by Filippenko (2004,
figure 2).

Using Matlab, Bob Day has run chi2 tests on the modified magnitude-
redshift relation for the Riess (2004) and Astier (2005) data sets
(figures 4, 5). For the Astier data he removed two outliers also removed
by Astier, leaving 115 data points. The best fit for the flat space
standard model has Wsppppp2.63 with chi^2ppppp113.5679, and for the
teleconnection model Wppppp1.06, with chi^2ppppp115.2. For the Riess data he
also removed outliers, leaving 154 data points. The best fit with the
standard model was Wppppp30 and chi^2ppppp 174.1. The best fit with the
teleconnection model was Wppppp1.15 with chi^2ppppp183.

5 Anomalous Pioneer Blue-shift
For some years the Pioneer spacecraft have been sending back Doppler
information interpreted as an anomalous acceleration toward the sun
(Anderson et al., 2002). No accepted explanation has been given for the
anomalous blue-shift, but if it were not observed it would be fatal to
this model. It is here seen as an optical effect due to expansion; wave
packets do not follow geodesics and there is disparity between the
solution of a wave function projected back in time from a final
measurement and the classical motion of a body. The disparity is removed
when the wave function collapses and coordinates are rescaled, but it
leads to an anomalous blue-shift in Doppler measurements of stellar
objects and the model predicts blue-shift simulating constant
acceleration toward the origin of coordinates, that is toward the
observer on Earth. This is a quantum effect; consistent with NASA's
findings, there is no corresponding classical acceleration and planetary
motions are unaffected. A future test is planned which will determine
whether the acceleration is toward the Sun, toward the Earth, in the
direction of motion of the craft, or along the spin axis (Nieto et. al.
2004). If the direction is not toward the Earth the test will falsify
this model.

Anderson remarks that the anomalous Pioneer blue-shift is equivalent to
an `acceleration in time' equal to the Hubble constant, but rejects
acceleration in time because, using conventional physics, it is
incompatible with ranging data. They elected to express their result in
the form of an equivalent classical acceleration,
a_pppppp8.74+-1.33x10^-8cms-2. A laboratory moving with respect to the
cosmic fluid uses locally Minkowski coordinates, which can be
transformed locally to comoving coordinates and extended globally to
coordinates with metric 3.2. Then the time coordinate obeys B3.4, and
exhibits acceleration with respect to proper time. The value of the
acceleration in time is H0/2 but blue-shift is doubled in 3.1. So the
resultant shift is the same. The equivalent acceleration, a_ppppppHc, is
consistent with recent determinations of Hubble's constant.

To gain an intuitive understanding of the Pioneer blue-shift consider a
wave packet for a particle of zero momentum and at a distance r from an
observer moving with the cosmic fluid at time t0, on a spacecraft in
empty space where gravity can be ignored. The coordinate space
displacement vector of the particle from the observer is a horizontal
arrow of constant length. By 2.1, the scaling distortions in quantum
coordinates are such that the actual displacement represented by the
coordinate space displacement vector is greater at time t0 than it was
at time tT0, by a factor a0^2/a^2. But the particle is also on a
geodesic moving with the cosmic fluid. So its actual displacement is
greater by a factor, a0/a. Then the motion of the wave packet and that
of the particle diverge, and the wave packet accelerates toward the
observer. Consistency is restored with the collapse of the wave function
and rescaling of coordinates. There is no classical acceleration but the
acceleration of the wave packet results in the anomalous blue-shift.

6 Flattening of Galaxies' Rotation Curves
The Pioneer blue-shift is present in the observation of distant
galaxies, and precisely accounts for flattening of galaxies' rotation
curves consistent with MOND, the phenomenological law found by Milgrom
(1994) which replaces the inverse square law of gravity with an inverse
law for accelerations a<<a_M. A review of MOND is given by Sanders &
McGough (2002). This is a second order effect, analogous to the
gravitational effect of the moon which produces the tides. The second
order pioneer effect is a Doppler shift equivalent to an inward
acceleration toward the galactic center and is observed as an apparent
modification to orbital velocity.

The anomaly is an optical effect arising from the treatment of redshift,
not a change to Newtonian dynamics (section 3.3) or evidence of cold
dark matter haloes. The accelerations of galaxies in clusters are in the
MONDian regime, and after revising the redshift-age relation; there is
no immediate evidence that CDM is necessary for galaxy evolution. The
MOND test is particularly important for several reasons. Firstly, data
fits have been given for over 100 galaxies and thousands of stars,
secondly, because cold dark matter does not give any explanation as to
why the precisely same acceleration law should be found in galaxies of
many sizes and types, thirdly, because there is no other empirical
evidence for CDM haloes, fourthly because there is no satisfactory
theory of CDM in particle physics, and finally because if galaxies'
rotation curves did not obey MOND it would refute this as a `CDM'
model.

A star, S, in a galaxy with centre, G, is subject to an acceleration due
to gravity toward G. Doppler shift due to orbital velocity is maximised
when S is on a diameter perpendicular to the Earth. In addition there is
an observed blue-shift equivalent to the Pioneer acceleration, g_rppppp-Hc,
toward the Earth. For ease of calculation project the galaxy into the
plane of its major axis and the Earth. Set up locally Minkowski
coordinates with an origin at G, with the y-axis directed toward the
Earth and with S on the x-axis at (x,0). The Pioneer blue-shift is
equivalent to an acceleration which can be resolved into a part
g_yppppp-g_r+O(x^2), and a part g_x toward G. g_x is independent of galactic
mass and would appear in Minkowski coordinates with an origin anywhere
in space. For a star on a diameter perpendicular to the Earth, the
Doppler shift due to g_x is equivalent to an orbital velocity v_P
measured relative to quantum coordinates, which are static with respect
to the cosmic fluid. v_P is due to expansion and depends on the position
of the star, but is independent of its velocity, so that Pioneer shift
due to v_P is simply added to Doppler shift due to motion.

Quantum coordinates with metric 3.2 are stretched in time by a factor of
2 and in the transverse direction by a factor of 1/2. So the blue-shift
corresponding to expansion from G is subject to a factor of 1/4.
Acceleration is the second time derivative of the transverse distance,
and requires a factor of 8 in these coordinates. So we find a net factor
of 32, giving a perceived acceleration,

g_xppppp -H0c/32 +O(x^2). 6.1

If the corresponding Doppler shift is interpreted as being due to the
motion of a body in orbit about G with orbital velocity then, to first
order

v_p^2/x ppppp H0c/32 or v_p ppppp sqrt(H0cx/32). 6.2

If the true orbital velocity of S due to gravity is vg then the observed
orbital velocity is

v ppppp v_g + v_P ppppp sqrt(GM/x) + sqrt(H0cx/32). 6.3

6.3 recognises that, since the alteration to redshift is an optical
effect, it is correct to add velocities, not accelerations as would be
the case for a dynamical law. Then the apparent acceleration toward G is

v^2/x ppppp GM/x^2 + sqrt(GMH0cx/8)/x + H0c/32. 6.4

The first term on the right hand side of 6.4 is acceleration due to
gravity. The last is simply the component of Pioneer acceleration toward
G, given by 6.1, and appears also in the absence of a source of gravity.
This leaves an apparent acceleration equivalent to a Doppler shift due
to velocity,

v^2 ppppp sqrt(GMH0c/8), 6.5

in precise agreement with MOND, the phenomenological law proposed by
Milgrom (1994) which retains Newton's square law for accelerations
xdotdot>>a_M for some constant a_M, but replaces it with

xdotdot ppppp sqrt(GMa_M) for xdotdot << a_M

and gives a good match with data. Thus a_MpppppH0c/8 and the best fit value
of a_M from observations on over a thousand stars is 1x10^-8cm s-2,
consistent with the emerging value of Hubble's constant and with the
value of Hc given by Pioneer.

7 Big Bang Nucleosynthesis and CBR
The square law applies when all the information about the initial state
is contained in the detected light, as in the observation of
astronomical bodies. The cosmological microwave background defines the
reference frame in which photons are emitted. This scales coordinates at
the time of the production of CMB photons and the usual linear red shift
law applies. The analyses of big bang nucleosynthesis and of decoupling
are unaltered, but the density of baryonic matter becomes
0.064<pppppW_Bh^2<ppppp0.096 after normalising Wcr to 1 (4.2). Thus baryonic
matter forms 10-20% of critical mass, and at an extreme, the ratio of
non-baryonic to baryonic matter need only be 4:1 for closure, within the
range of values which might be accounted for by a massive neutrino.

The concordance model is supported by the integrated Sachs-Wolfe effect
(Afshordi, Loh & Strauss; 2004; Boughn & Crittendon, 2004; Fosalba et
al., 2003; Nolta et al., 2004; Scranton et al., 2004) using evidence
from the Two-Degree Field Galaxy Redshift Survey (2dFGRS; Peacock et al.
2001; Percival et al., 2001; Efstathiou, 2002), and from the Wilkinson
Microwave Anisotropy Probe (WMAP; Spergal, 2003, and references cited
therein). In practice these measurements determine cosmological
parameters rather than test consistency, and they depend on the
distance-redshift relation. Acceleration depends only on distance and
time, so that, if the standard model is consistent, a change in the
distance-redshift relation can be expected to give a consistent change
in the deceleration parameter in different tests. It is to be expected
that corresponds to in the teleconnection model whether it is
determined from Supernova or from WMAP and 2dGFS.

The first order analysis of WMAP appears unchanged in the teleconnection
model, as we expect isotropy and a gaussian random distribution. However
Spergal comments on discrepancies in the WMAP data on both the largest
and smallest scales, and Copi et al (2005) report on unexplained
alignments in the data. It is not presently possible to say whether
these are caused by higher order corrections in the analysis of data;
for example it may be necessary to take account of pioneer blue-shift
when removing foreground contamination.

Appendix A: Hubble Diagrams for Type Ia Supernovae

Appendix B: Mathematical Description of the Teleconnection

Regards

--
Charles Francis
Please reply by name

Charles Francis

[Moderator's note: Charles, please contact me via email. -P.H.]


Thus spake Charles Francis <charles@CF.wanadoo.co.uk>
>Thus spake Charles Francis <charles@CF.wanadoo.co.uk>
>>
>>
>>4 Cosmological Parameters
>>The interpretation of redshift alters cosmological parameters but
>>otherwise leave the classical equations of general relativity unchanged.
>>Friedmann's equation is
>>
>> [as usual]. 4.1
>>
>>Normalising so that W=1 is critical density we define
>>
>> W=..., W_R=... and W_L=..., 4.2
>>
>>where k=-1,0,1 (quadruple the standard values, so that W=1 is critical
>>density). Then Friedmann's equation, 4.1 is,
>>
>> adot/a = H0/2 (W(1+z)^3/2 + W_R(1+z) + W_L)^1/2 4.3
>>
>>[standard form has redshift factors squared and no /2], requiring that W
>>+ W_R + W_L =1. [as standard]
>>
>>Definition: Let the deceleration parameter be q=-4a adotdot/aH^2
>>
>>Differentiate Friedmann's equation to find the acceleration equation,
>>
>> [similarly similar to standard]. 4.4
>>
>>Let H0=H(t0), a=a(t0) and q0=q(t0). From 4.2 and 4.4,
>>
>> q=W/2 -W_L 4.5
>>
>>[standard]
>>
>[continued]
>Angular size distance is calculated as in the standard theory, but
>rescaling of coordinates introduces a factor of a0/a(t)=sqrt(1+z), which
>adds 2.5log(1+z) to magnitude. To first order in z the magnitude-
>redshift relation (c.f. Misner Thorne Wheeler 1973, eq. 39.35b) becomes
>
> m-M ~ 5logz + 1.086(2-q)z 4.6
>
[I don't know why this has lost most of the post, but it's the second
time it's happened, so here's hoping it will all come through this time]

Then if qs is the deceleration parameter for the standard model q0~qs+1.
For a flat space model, W=Ws+2/3. If L=0, W=3Ws. In this case an
estimate of the error may be made by combining the systematic and
statistical errors given by Astier, and multiplying by 3. For critical
density, W=1, WLs=0, 4.6 is identical to first order, to the magnitude-
redshift relation with Ws=0.33, WLs=0.67. The plot for this model has
been superimposed in green on residual Hubble diagrams by adding
2.5log(1+z) to the standard plot (figures 1, 2, 3). These show that, for
Z>0.4 the fit is closer to Ws=0.26, WLs=0.74, the best fit for the
Supernova Legacy Survey first year data (Astier et al, 2005, figure 1),
than it is to the Ws=0.35, WLs=0.65 model given by Filippenko (2004,
figure 2).

Using Matlab, Bob Day has run chi2 tests on the modified magnitude-
redshift relation for the Riess (2004) and Astier (2005) data sets
(figures 4, 5). For the Astier data he removed two outliers also removed
by Astier, leaving 115 data points. The best fit for the flat space
standard model has Ws=2.63 with chi^2=113.5679, and for the
teleconnection model W=1.06, with chi^2=115.2. For the Riess data he
also removed outliers, leaving 154 data points. The best fit with the
standard model was W=30 and chi^2= 174.1. The best fit with the
teleconnection model was W=1.15 with chi^2=183.

5 Anomalous Pioneer Blue-shift
For some years the Pioneer spacecraft have been sending back Doppler
information interpreted as an anomalous acceleration toward the sun
(Anderson et al., 2002). No accepted explanation has been given for the
anomalous blue-shift, but if it were not observed it would be fatal to
this model. It is here seen as an optical effect due to expansion; wave
packets do not follow geodesics and there is disparity between the
solution of a wave function projected back in time from a final
measurement and the classical motion of a body. The disparity is removed
when the wave function collapses and coordinates are rescaled, but it
leads to an anomalous blue-shift in Doppler measurements of stellar
objects and the model predicts blue-shift simulating constant
acceleration toward the origin of coordinates, that is toward the
observer on Earth. This is a quantum effect; consistent with NASA's
findings, there is no corresponding classical acceleration and planetary
motions are unaffected. A future test is planned which will determine
whether the acceleration is toward the Sun, toward the Earth, in the
direction of motion of the craft, or along the spin axis (Nieto et. al.
2004). If the direction is not toward the Earth the test will falsify
this model.

Anderson remarks that the anomalous Pioneer blue-shift is equivalent to
an `acceleration in time' equal to the Hubble constant, but rejects
acceleration in time because, using conventional physics, it is
incompatible with ranging data. They elected to express their result in
the form of an equivalent classical acceleration,
a_=p8.74+-1.33x10^-8cms-2. A laboratory moving with respect to the
cosmic fluid uses locally Minkowski coordinates, which can be
transformed locally to comoving coordinates and extended globally to
coordinates with metric 3.2. Then the time coordinate obeys B3.4, and
exhibits acceleration with respect to proper time. The value of the
acceleration in time is H0/2 but blue-shift is doubled in 3.1. So the
resultant shift is the same. The equivalent acceleration, a_=pHc, is
consistent with recent determinations of Hubble's constant.

To gain an intuitive understanding of the Pioneer blue-shift consider a
wave packet for a particle of zero momentum and at a distance r from an
observer moving with the cosmic fluid at time t0, on a spacecraft in
empty space where gravity can be ignored. The coordinate space
displacement vector of the particle from the observer is a horizontal
arrow of constant length. By 2.1, the scaling distortions in quantum
coordinates are such that the actual displacement represented by the
coordinate space displacement vector is greater at time t0 than it was
at time tT0, by a factor a0^2/a^2. But the particle is also on a
geodesic moving with the cosmic fluid. So its actual displacement is
greater by a factor, a0/a. Then the motion of the wave packet and that
of the particle diverge, and the wave packet accelerates toward the
observer. Consistency is restored with the collapse of the wave function
and rescaling of coordinates. There is no classical acceleration but the
acceleration of the wave packet results in the anomalous blue-shift.

6 Flattening of Galaxies' Rotation Curves
The Pioneer blue-shift is present in the observation of distant
galaxies, and precisely accounts for flattening of galaxies' rotation
curves consistent with MOND, the phenomenological law found by Milgrom
(1994) which replaces the inverse square law of gravity with an inverse
law for accelerations a<<a_M. A review of MOND is given by Sanders &
McGough (2002). This is a second order effect, analogous to the
gravitational effect of the moon which produces the tides. The second
order pioneer effect is a Doppler shift equivalent to an inward
acceleration toward the galactic center and is observed as an apparent
modification to orbital velocity.

The anomaly is an optical effect arising from the treatment of redshift,
not a change to Newtonian dynamics (section 3.3) or evidence of cold
dark matter haloes. The accelerations of galaxies in clusters are in the
MONDian regime, and after revising the redshift-age relation; there is
no immediate evidence that CDM is necessary for galaxy evolution. The
MOND test is particularly important for several reasons. Firstly, data
fits have been given for over 100 galaxies and thousands of stars,
secondly, because cold dark matter does not give any explanation as to
why the precisely same acceleration law should be found in galaxies of
many sizes and types, thirdly, because there is no other empirical
evidence for CDM haloes, fourthly because there is no satisfactory
theory of CDM in particle physics, and finally because if galaxies'
rotation curves did not obey MOND it would refute this as a `CDM'
model.

A star, S, in a galaxy with centre, G, is subject to an acceleration due
to gravity toward G. Doppler shift due to orbital velocity is maximised
when S is on a diameter perpendicular to the Earth. In addition there is
an observed blue-shift equivalent to the Pioneer acceleration, g_r=-Hc,
toward the Earth. For ease of calculation project the galaxy into the
plane of its major axis and the Earth. Set up locally Minkowski
coordinates with an origin at G, with the y-axis directed toward the
Earth and with S on the x-axis at (x,0). The Pioneer blue-shift is
equivalent to an acceleration which can be resolved into a part
g_y=-g_r+O(x^2), and a part g_x toward G. g_x is independent of galactic
mass and would appear in Minkowski coordinates with an origin anywhere
in space. For a star on a diameter perpendicular to the Earth, the
Doppler shift due to g_x is equivalent to an orbital velocity v_P
measured relative to quantum coordinates, which are static with respect
to the cosmic fluid. v_P is due to expansion and depends on the position
of the star, but is independent of its velocity, so that Pioneer shift
due to v_P is simply added to Doppler shift due to motion.

Quantum coordinates with metric 3.2 are stretched in time by a factor of
2 and in the transverse direction by a factor of 1/2. So the blue-shift
corresponding to expansion from G is subject to a factor of 1/4.
Acceleration is the second time derivative of the transverse distance,
and requires a factor of 8 in these coordinates. So we find a net factor
of 32, giving a perceived acceleration,

g_x= -H0c/32 +O(x^2). 6.1

If the corresponding Doppler shift is interpreted as being due to the
motion of a body in orbit about G with orbital velocity then, to first
order

v_p^2/x = H0c/32 or v_p = sqrt(H0cx/32). 6.2

If the true orbital velocity of S due to gravity is vg then the observed
orbital velocity is

v = v_g + v_P = sqrt(GM/x) + sqrt(H0cx/32). 6.3

6.3 recognises that, since the alteration to redshift is an optical
effect, it is correct to add velocities, not accelerations as would be
the case for a dynamical law. Then the apparent acceleration toward G is

v^2/x = GM/x^2 + sqrt(GMH0cx/8)/x + H0c/32. 6.4

The first term on the right hand side of 6.4 is acceleration due to
gravity. The last is simply the component of Pioneer acceleration toward
G, given by 6.1, and appears also in the absence of a source of gravity.
This leaves an apparent acceleration equivalent to a Doppler shift due
to velocity,

v^2 = sqrt(GMH0c/8), 6.5

in precise agreement with MOND, the phenomenological law proposed by
Milgrom (1994) which retains Newton's square law for accelerations
xdotdot>>a_M for some constant a_M, but replaces it with

xdotdot = sqrt(GMa_M) for xdotdot << a_M

and gives a good match with data. Thus a_M=H0c/8 and the best fit value
of a_M from observations on over a thousand stars is 1x10^-8cm s-2,
consistent with the emerging value of Hubble's constant and with the
value of Hc given by Pioneer.

7 Big Bang Nucleosynthesis and CBR
The square law applies when all the information about the initial state
is contained in the detected light, as in the observation of
astronomical bodies. The cosmological microwave background defines the
reference frame in which photons are emitted. This scales coordinates at
the time of the production of CMB photons and the usual linear red shift
law applies. The analyses of big bang nucleosynthesis and of decoupling
are unaltered, but the density of baryonic matter becomes
0.064<=W_Bh^2<=0.096 after normalising Wcr to 1 (4.2). Thus baryonic
matter forms 10-20% of critical mass, and at an extreme, the ratio of
non-baryonic to baryonic matter need only be 4:1 for closure, within the
range of values which might be accounted for by a massive neutrino.

The concordance model is supported by the integrated Sachs-Wolfe effect
(Afshordi, Loh & Strauss; 2004; Boughn & Crittendon, 2004; Fosalba et
al., 2003; Nolta et al., 2004; Scranton et al., 2004) using evidence
from the Two-Degree Field Galaxy Redshift Survey (2dFGRS; Peacock et al.
2001; Percival et al., 2001; Efstathiou, 2002), and from the Wilkinson
Microwave Anisotropy Probe (WMAP; Spergal, 2003, and references cited
therein). In practice these measurements determine cosmological
parameters rather than test consistency, and they depend on the
distance-redshift relation. Acceleration depends only on distance and
time, so that, if the standard model is consistent, a change in the
distance-redshift relation can be expected to give a consistent change
in the deceleration parameter in different tests. It is to be expected
that corresponds to in the teleconnection model whether it is
determined from Supernova or from WMAP and 2dGFS.

The first order analysis of WMAP appears unchanged in the teleconnection
model, as we expect isotropy and a gaussian random distribution. However
Spergal comments on discrepancies in the WMAP data on both the largest
and smallest scales, and Copi et al (2005) report on unexplained
alignments in the data. It is not presently possible to say whether
these are caused by higher order corrections in the analysis of data;
for example it may be necessary to take account of pioneer blue-shift
when removing foreground contamination.

Appendix A: Hubble Diagrams for Type Ia Supernovae

Appendix B: Mathematical Description of the Teleconnection

Regards

--
Charles Francis
Please reply by name