The document discusses separating gravitational redshift from cosmic expansion redshift in observed cosmic microwave background radiation. It presents calculations of curvature ratios at different points in the early universe, such as at electroweak symmetry breaking and at the surface of last scattering. Taking the curvature ratio at these two points and using observed CMB redshift of 1091, the analysis estimates gravitational redshift would be around 2-3, with most of the observed redshift coming from cosmic expansion. This has implications for the energy density and relativistic nature of the early universe.

1. My new article on Electro-Weak redshift.
1

2. Can Gravitational Redshift and Cosmic Expansion Redshift
Be Separated?
Douglas Leadenham
March 30, 2016
1 Introduction
In a 2005 seminar for high school teachers at SLAC National Accelerator Laboratory a
teacher asked the moderator how much of observed cosmic microwave background redshift
is gravitational and how much results from recessional velocity from us, the observers. She
responded that the answer depends on whether the early universe was ultra relativistic or
not, and that question can’t be answered at present.
1.1 Motivation
Years later in 2014 an astrophysics researcher also claimed that the two factors can’t
be separated. Now that the Higgs field and the field strength of electro-weak symmetry
breaking are known the two factors may be separable.
2 Higgs Field
Section 7.4, Eq.7.14, of 21st Century Physics1 provides the Higgs field:
EH =
ln 2
4π
kF EP (1)
In this simple expression, kF is the Fermi level of the degenerate primordial fireball and
EP is the Planck energy.
1
Douglas Leadenham, Topics in 21st
Century Physics - The Universe as Presently Understood, DJLe-
Books, 2016
2

3. 2.1 Field Curvature
From basic physics one knows the energy density of Earth’s gravitational field.
ug =
g2
8πG
(2)
Section 8.7.1, Eq. 8.25, of the same book gives the corresponding energy density of the
fireball at electro-weak (E-W) symmetry breaking:
uw =
3 (ln 2)2
k2
F c2
(4π)3
G 2
(3)
From this the functional equivalent of the E-W field strength is
g =
√
6 ln 2
4π
kF c
(4)
Simple dimensional analysis confirms that this g represents an acceleration like Earth’s g,
but a kinematical formula is not defined before de Sitter space came into existence. Divide
(4) by c2 to convert it to an analog of more general curvature. This can form a ratio with
the effective curvature of the whole universe at its logical beginning.
g
c2
=
√
6 ln 2
4π
kF
c
= 59 (5)
This curvature is measured in meters−1. For comparison and perspective the curvature of
g in (2) is 9.8
c2 meters−1.
2.2 Universal Curvature
Start with a spherical universe with a horizon:
2Gµ
c2
=
2GMu
Ruc2
= 1 (6)
Here Mu is the effective mass of the universe and Ru is the Hubble length. In the inflation-
ary model, now confirmed by modulation of the cosmic microwave background, although
some skeptics attach more credence to the overall signal, there must be more of the universe
beyond the horizon, because inflation is superluminal, assuming that the speed of light was
constant at the beginning. The speed of light may have been much greater then, but we
can’t be sure of that. Section 8.5, Eq. 8.19, gave the Hubble constant:
H0 =
mec
π exp 1
e2c
(7)
3

4. The result of Chapter 3, obtained by a linear optimization on galaxy rotation, gave the
tension at GUT symmetry breaking:
Gµ
c2
= 4πµ0α =
µ2
0e2c
(8)
So that the curvature factor we seek for the universe is
H0
c
=
4α
exp 1
me
µ0
(9)
However, this is for the observable universe. There is an unknown part that must be ac-
counted for. We will use an adjustable factor n, as follows below. Use the analogy of
Newton’s universal law of gravitation and divide by c2 to convert the kinematical formu-
lation to one of curvature.
2GMu
c2R2
u
=
1
Ru
2Gµ
c2
(10)
Substitute for the Hubble length and obtain
2GMu
c2R2
u
=
4α
exp 1
me
µ0
2Gµ
c2
(11)
The last factor on the right, 2Gµ
c2 , represents the whole universe, including the unobservable
part beyond the horizon. The left hand side represents the curvature of this mostly unseen
universe, so the right hand side must include the adjustable factor n. Express (10) this
way:
2GMu
c2R2
u
=
4α
exp 1
me
µ0
2Gµ
c2
=
4α
exp 1
me
µ0
2G
c2
mP
lP
(12)
The ratio of Planck mass to Planck length is by definition c2
G . That leaves 2G
c2 , which is 1
µ,
the inverse of a curvature, less than an unknown, higher-dimensional universal curvature
µu by a dimensionless adjustable factor n. If we try n = 6, (11) evaluates to 63 meters−1,
for comparison to (5). Rewrite (11) as:
2GMu
c2R2
u
=
4α
exp 1
me
µ0
1
µ
c2
G
=
4α
exp 1
me
µ0
1
µu
n
c2
G
(13)
The dimensionless ratio of curvatures becomes:
√
6 ln 2
4π
kF
c
4α
exp 1
me
µ0
nc2
G
=
exp 1 ln 2
√
6
8πα
kF µ0G
n mec3
(14)
4

5. 3 Redshift
At the very beginning the general relativistic gamma-factor is
γ =
1
1 −
GEp
lpc4
108
(15)
With knowledge that redshift results from frequency reduction due to time dilation, replace
the hypothetical
GEp
lpc4 with (14). Vary n to see how redshift z responds.
zg + 1 =
1
1 − exp 1 ln 2
√
6
8πα
kF µ0G
n mec3
(16)
Observed redshift z is the product of the gravitational field redshift zg with the cosmic
expansion redshift zc as
z + 1 = (zg + 1)(zc + 1) (17)
The Planck Law of black body radiation is
I(λ, T) =
2πhc2
λ5(e
hc
λkBT
− 1)
(18)
The black body radiation intensity has a maximum at hc
λkBT = 2.82144. This λ is the
wavelength of the cosmic background radiation that is red shifted by a factor of 1091,
according to recent measurement. To preserve symmetry we will put back into (14) the
factor 2 that goes with G.
exp 1 ln 2
√
6
8πα
kF µ02G
n mec3
< 1 (19)
Now the parameter n can be given integer values from 11 to 14 to show how (16) varies
in response. Taking n = 11 makes γ imaginary. Take n = 12 and zg = 3.02; n = 13 gives
zg = 2, and n = 14 gives zg = 1. The last one is unreasonably small, but zg = 3 is certainly
understandable, as is zg = 2.
At this point one could reason that the integers just chosen may have something to do with
the number of dimensions of the universe of three mutually orthogonal sectors. It is known
that there must be at least 11 dimensions, so the integer 12 possibly is significant.
If we take the composite redshift to be 1091, then the cosmic expansion redshift factor
then is zc = 272 if zg = 3, and zc = 363 if zg = 2. These are reasonable results. However,
there is a problem.
5

6. 4 The Problem
Here is the situation: the field strength of (5) is that of the primordial universe at E-W
symmetry breaking, and it is ratioed to the field at the very beginning 13.8 billion years
ago, while the observed CMB radiation originated at the epoch of last scattering when
neutral hydrogen could form. The redshift measured now is relative to the emitted CMB
radiation. These two pairs of events occurred at very different times and circumstances,
most notably the ambient field strength. The analysis presented here may be correct in
principle, but while the fields of the analysis can be right, the CMB radiation did not
originate with them. While the present day universal average field strength is known, the
relevant field at last scattering is not. The challenge is to find it.
5 Resolution
Here we will redo the redshift calculation of §2.2 for the relevant fields. Energy density
uCMB at the surface of last scattering is the equivalent energy of the observable universe
divided by the volume of the (Euclidean flat) universe at the age of the universe then. This
time is 375,000 years, because the surface of last scattering is more like a dense ball of
water than a transparent balloon. We will call that time tCMB. We have
uCMB =
3Ruc
8πGt3
CMB
=
1
8πG
g2
CMB (20)
gCMB
c2
=
3Ru
c3t3
CMB
(21)
Ru =
exp 1µ0
4αme
(22)
The corresponding energy density of the present-day universe is the same universal equiv-
alent energy divided by the equivalent volume of the Euclidean universe.
uNOW =
3c4
8πGR2
u
(23)
gNOW
c2
=
√
3
Ru
(24)
The dimensionless ratio desired to substitute into (15) is
c3t3
CMB
R3
u
(25)
6

7. This evaluates to 1.48 × 10−7. This looks close to the pattern, 1.16 × 10−7, found in §3.6
of the cited e-book, see §8 below. Just for curiosity, put in an age of the CMB that gives
that figure. It is 323,000 years. This age falls into the inflationary period prior to the
surface of last scattering where the CMB radiation originates. The CMB corresponds to a
temperature 3000 K for which the radiation does not ionize hydrogen. The earlier time
corresponds to the time when stable electrons and quarks condensed from a radiation field
of temperature 250 million K, left over after anti-particles annihilated or aggregated into
nascent black holes. This modeling does not conflict with our limited knowledge of the
early universe, so we can be satisfied with our estimates of both components of redshift
z = 1091.
6 Conclusion
The redshift factor zg = 3 of §3 above is correct in principle, but it is unobservable. What
it tells us anyway is that redshift due to field strength is quite small compared to redshift
resulting from recessional velocity of cosmic inflation. Redshift of field strength results
from a large field strength difference between two points of measurement. In theory we see
that this difference between E-W symmetry breaking and the very beginning is modest.
That means that the universe was modestly relativistic at early times, not ultra-relativistic.
That the redshift of the CMB is mostly due to cosmic inflation squares with recent CMB
polarization results that the universe began with exponential inflation showing gravitational
wave effects from an early epoch. The 1091 CMB redshift factor gives a β = 0.9999996
factor for the recessional velocity of the neutral hydrogen of the young universe of 375,000
years.
The most informative information to be gleaned from this analysis could be the ratio of
curvature at E-W symmetry breaking to curvature calculated for the CMB. This ratio is
59/1.16 × 10−7, 500 million times greater. If curvature of the nascent universe at the time
of last scattering is modest, then at the time of E-W symmetry breaking it would have been
modestly relativistic with β = 0.205. Recent measurement of B-mode polarization effects
of gravitational radiation show a strong signal in the CMB background where a weak one
was expected.2 Maybe we know more about the early universe than we thought.
2
BICEP2 Collaboration, Bicep 2 I: Detection of B-mode polarization at degree angular scales. 18 Mar
2014. Phys. Rev. Lett. 112, 241101 (2014) arXiv:1403.3985v2 [astro-ph.CO]
7

8. 7 Implications
Here is the widely publicized graphic portrayal of the growth of the universe from NASA:
ImageCredit : NASA/WMAPScienceTeam
This graphic shows the universe as it evolved from the Big Bang to now, the 13.77-billion-
year-long history of our universe.
The present-day gentle Dark Energy Accelerated Expansion shows as the flaring horn at
the right end of the graphic. The exponential inflation resulting from GUT symmetry
breaking in the instant of Quantum Fluctuations is at the extreme left end. The slowly
expanding part between the ends is the matter dominated phase that slows expansion
after the inflaton field shuts off. Present-day acceleration would shut off also, if all charged
particles paired up and fell into the ground state, and the Higgs field shuts off.
Exponential inflation is described in Chapter 3 of the cited book, and dyons, the first
particles of quantum fluctuations, pair up to make quadrons. In the fleeting moment that
they are free, they exist in an enormously strong field analogous to the Higgs field of our era.
We call this field the inflaton, in that it expands the universe in the era labeled Inflation in
8

9. the graphic. Formation of quadrons neutralizes the dyons except for their electric charges,
as described in Chapter 5, and that shuts down the inflaton field when quadrons fall into
stable states. An acceleration is seen now as an effect of the Higgs field of luminous matter,
described in Chapter 8, that gives the observed universal acceleration that will continue
indefinitely. Eons from now, if the universe cooled off completely and all matter fell into
the ground state, the impetus of expansion would shut off, and the remaining matter would
decelerate gravitationally until it heated up again, restoring the Higgs field.
Food for thought: the Higgs field is known to have a higher energy component, and hints
of it have been observed in experiments at the Large Hadron Collider. Is this higher energy
field the remnant of the inflaton?
8 §3.6 of the Book; Dimensionless Magnetic Constant µ0
This constant first appeared in 19th century laboratory work with the field of a current.
In textbooks it shows up as below:
dB =
µ0
4π
ids sin θ
r2
−→
B is the magnetic field in Teslas at a point P, i is the current in Amperes, s is the arc
length in meters, θ is the angle between d−→s and the radius vector −→r from ds to P, with
magnitude r also in meters. In this definition µ0 ≡ 4π × 10−7T-m/A. In current lists of
constants it is given as µ0 ≡ 4π × 10−7N/A2, akin to a pressure. These classical physics
definitions are very far from the quantum mechanical general relativistic conditions at the
founding of the universe being discussed here. In general relativity it is customary to
give physical observables in units of an equivalent length. For example the solar mass
M = 1.988435Ö1030kg is usually expressed as a length RS = 2.96 km, the radius of a
black hole enclosing a solar mass. Likewise, a single kilogram of mass is expressible as
2G
c2 = 1.49×10−27m. Now return to the regression result 4πµ0α = 1.152×10−7N/A2. This
SI unit is equivalent to kg−m/s2
(C
s )
2 = kg−m
C2 because µ0 has units of N/A2. The mass part can
be converted to a length equivalent by multiplying by 2G
c2 . Now observe that
µ0 = 4π × 10−7 kg − m
C2
×
2G
c2
m
kg
=
8πG
c2
× 10−7 m2
C2
=
2G
c2
µ0
m2
C2
(26)
If we express a single charge q expressed in Coulombs C as
q ⇒ q
2G
c2
µ0 = 4.32 × 10−17 m
C
× q(C) (27)
9

10. then µ0 is truly a dimensionless constant. This is a new general relativity expression for
electric charge that we should call the Coulomb length to go with Planck length and Planck
electric charge set out in §3.2. For example, the fundamental electric charge e in general
relativity can be expressed as
e = 4.32 × 10−17 m
C
× −1.6021766 × 10−19
C = −6.92 × 10−36
m (28)
This carries a profound implication, namely that the founding of the universe began with
electromagnetism. There is one force only: electromagnetism, that warps space-time giving
gravity and the electromagnetic field. Every observable in physics results from these field
effects. This now begs the question: What is mass?
Related this question is the often-quoted electron charge-to-mass ratio: 1.758820024×1011
C/kg, called a ratio in the sense of “amount of this per unit of that”. Now that it is
possible to express both charge and mass in consistent units of measurement, the ratio is
better understood as a true dimensionless quotient.
e
me
= (1.7588×1011
C/kg)×
2G
c2
µ0m/C/(
2G
c2
m/kg) = (1.7588×1011
C/kg)
µ0c2
2G
= 5.116×1021
(29)
Note that this is over 10 billion times larger that the traditional numerical value. It is no
wonder that J.J. Thomson and many physics students after him were able to move the
electrons so easily with electric and magnetic fields in their lab experiments.[?]
There is more. If in fact there is only the electromagnetic force in nature, then the gravi-
tational constant G is not as fundamental as thought and could be expressible in terms of
electromagnetic constants. Invert the last equation, take the cube root and multiply by 2.
This may be contrived, but notice the result:
2
3 me
e
2G
µ0c2
= 1.16 × 10−7
. (30)
An obvious pattern has emerged. It is thought that electric charge and mass involve
different dimensions of 11-dimensional space-time, so this may suggest that mass involves
3 times as many of them as charge. We use a value of G obtained by laboratory experiment
or observational astrophysics that is true today. We applied it in the case of the electron
charge-to-mass ratio to get the measurement dimensions right. If the observed pattern has
any validity, we can obtain a value of G that was true at the founding of the universe with
GUT symmetry breaking. That value of G need not equal the modern observed value, and
10

11. there have been many speculative articles published that suggest that G has changed over
the 13.8 billion years of the universe’s existence. Let us see what we get from this.
2
3 me
e
2G
µ0c2
= 4πµ0α (31)
Isolating our hypothetical old value of G on the lhs, we get
GGUT = µ0c2 e2
m2
e
32π6
µ6
0α6
. (32)
This gravitational constant is proportional to the 7th power of µ0 and the 6th power of
the fine structure constant α. Even minute changes in either of them in 13.8 billion years
would affect the gravitational constant profoundly, but as follows, the situation is much
simpler.
Anyway we get
GGUT = 6.39138 × 10−11
(33)
as compared to today’s value, G = 6.67408×10−11m3/kg−s2, as reported by NIST. This is
4.2% smaller than the presently accepted value, not unreasonable given the long-speculated
possible change in G from the founding of the universe.[?]
Why would G be different then? Look at the denominator of eq. 3.26. You see the
electron’s rest mass, but at symmetry breaking it is certainly not at rest. We must replace
that mass-energy by its special relativity value p2c2 + m2
ec4, at 2.1% larger than the rest
value. This momentum −→p is directed radially and is not random as if thermal, so we
would use 1
2kBT instead of 3
2kBT to calculate a temperature for the kinetic energy. We
get 250 million Kelvin, also reasonable. Modify eq. 3.26 with a weak correction factor as
follows:
GGUT =
G
(1 + f (u))2 = µ0c2 e2
m2
e (1 + f (u))2 32π6
µ6
0α6
. (34)
In the calculation (1 + f (u))2
= 1.042, and f (u) = 0.021. This looks like special relativity
with (1 + f (u))2
= 1.042 = 1
1−v2
c2
= γ2 and v
c = 0.205, reasonable once more. The weak
correction runs from 1 to 1.021 as electron energy runs from laboratory temperature to
250 million K. By this reasoning, any experiment to measure G must take into account the
motion of the particles in the apparatus, if they are at all relativistic.
11

12. See what this principle means for the gravitational field of a moving electron ei toward
another, stationary electron ej at an inter-electron distance rij:
Vij = −
G
γ2
i
γimeme
rij
γi
= −
Gm2
e
rij
(35)
There is no change from Newton’s Law even though the gravitational constant is affected
noticeably. A possible test of such a field effect as this could be made near the beam of
the Large Hadron Collider. Higher energy of newly formed electrons at symmetry breaking
reduces the measured G, which could suggest that any or all ambient energy affects the
measurement. Could precision measurements of G be made at varying distances from
the LHC beam, both parallel to and perpendicular to the beam in order to estimate this
hypothesized weak effect?
12