Nikita Lovyagin et al. (2022) have plotted angular diameters of galaxy cores versus the redshift. The red data is from the James Webb space telescope:
There is a serious discrepancy in the observed data from the ΛCDM model. Lovyagin et al. measured the angular diameters from publicly available data. Are their measurements correct?
In the Milne cosmological model, the scale factor a of the universe increases at a constant speed. What is the plot like for the Milne model? Linear?
Density of early galaxies at high redshifts z
The angular diameter of a single galaxy in a photograph is a fuzzy concept. Maybe galaxy cores were much smaller at z = 10 than they are now, and that is why they appear to have a surprisingly small diameter in the photograph?
What about looking at the angular distance of two adjacent galaxies? If galaxies were born all at the same time in the distant past, we can calculate how many of them are at a certain value z in a square degree of the sky, for different cosmological models.
The diagram above suggests that James Webb might see 100X the galaxies at z = 10, compared to what ΛCDM predicts.
Stacy S. McGaugh (2024) shows some data about the observed density on massive galaxies at a high redshift z:
On the horizontal axis we have the rest-frame ultraviolet absolute magnitude MU of the galaxy. Blue denotes z = 9, red z = 14. We see that at z = 14, there is 100X the predicted density of galaxies. However, for z = 9, the discrepancy is only 3X.
Dark energy and the expansion rate
The second Friedmann equation says:
The "dark energy density" is equal to
-p
where p is the negative pressure. Suppose that the density of matter (excluding dark energy) is m. Then
ρ = m + -p / c².
Let us solve for a zero acceleration:
a'' = 0
<=>
ρ + 3 p / c² = 0
<=>
m - p / c² + 3 p / c² = 0
<=>
-p = 1/2 m c².
That is, the dark energy density -p is a half of the matter energy density. The "fraction" of dark energy is defined
fDE = -p / (ρ c²).
If the fraction is 1/3, then there is no acceleration in the expansion of the universe – it is like the Milne model.
The distance versus the redshift z in the Milne model
A G
light galaxy
o ~~~~~~~~~~~~~~~~ O
|
/\
observer
Let us determine the redshift of light in the Milne model, versus the apparent angular diameter of the galaxy sending that light.
It is like the balloon cosmological model, where the balloon grows at a constant speed.
If the distance between some points A and G grows faster than light, then there is a "horizon" between A and G.
A G
^ x • ●
| <------- S ------->
| coordinate
---------> y distance
Metric. Let us model the uniform stretching of space in the x, y plane. Galaxies and observers stay at fixed coordinates. It is the spatial metric which stretches. The flat spatial metric g stretches linearly with the time t:
g = a₀² t² I,
where I is the identity matrix and a₀ is a constant. Note that the stretching factor is the square root of an element in g. The scale factor a of the universe is
a(t) = a₀ t.
A photon moves along a straight line in the x, y plane. We assume that the speed of light, c, is constant (in the metric). The coordinate speed of the photon slows down as time passes.
Angular diameter. Let S be the coordinate distance between the observer A and the galaxy G. The proper diameter d₀ of G stays constant. The coordinate diameter of G at a time t is
D(t) = d₀ / a(t)
= d₀ / (a₀ t).
The angular diameter of the galaxy G, as seen by the observer A is
A(t) = D(t₀) / S,
where t₀ is the time at which the light observed by A left the galaxy G. The proper distance s of A and G at that time was
s = S a₀ t₀.
Then
A(t) = d₀ / (a₀ t₀) * (a₀ t₀) / s
= d₀ / s.
That is, we can determine the angle A(t) either by comparing coordinate distances or proper distances, of course.
Coordinate speed of light. The proper speed of light in the metric is c. The coordinate speed C(t) of light is
C(t) = c / a(t).
Redshift. We assume that the wavelength of a wave packet stretches as the spatial metric stretches. If the redshift is z, then a wavelength λ has stretched to
λ(t) = (1 + z) λ.
We have
z = a(t) - 1.
Redshift z as the light reaches the observer. We have to integrate, in order to obtain the time t when the light from an early galaxy G reaches our observer A. The light departs at a time t₀. The coordinate distance traveled by the light is
t
S = ∫ c / (a₀ t) * dt
t₀
= c / a₀ * (ln(t) - ln(t₀))
<=>
t = exp( a₀ / c * S + ln(t₀) )
= t₀ exp(a₀ / c * S).
The age of the universe. The current value of the Hubble "constant" is 73 km/s per megaparsec. A megaparsec is
3.09 * 10²² meters.
The scale factor of the universe grows by
73,000 / (3.09 * 10²²)
= 2.36 * 10⁻¹⁸ * 1 / second
= a₀.
The age of the universe is
1 / a₀ = 4.2 * 10¹⁷ seconds
= 13.3 billion years.
Proper distance of the last scattering of CMB. Let us have a photon of the cosmic microwave background which we observe right now. Let us determine the proper distance of the photon to us at the time when the photon was scattered the last time.
The density of the matter was the right one for the last scattering when the scale factor was 1 / 1,100 'th of the current one. Then we have
t₀ = 3.8 * 10¹⁴ seconds
= 12 million years.
From
1,100
= t / t₀
= exp(a₀ / c * S)
we get
ln(1,100) = a₀ / c * s / (a₀ t₀)
<=>
s = ln(1,100) * c t₀.
= 7 c t₀
= 84 million light-years.
The value for ΛCDM is 30 million light-years.
The age of the universe at the last scattering is 12 million years in the Milne model, but only 380,000 years in ΛCDM.
Baryon acoustic oscillations
The CMB angular spectrum matches the baryon acoustic oscillations for a 380,000 year old universe in ΛCDM. Could it be that they also match Milne?
The "sound horizon" determines the size scale of the acoustic oscillations. How far can a plasma sound wave reach in ΛCDM versus Milne?
Edward L. Wright (2014) writes that the most of the mass of the plasma is in photons, which means that the sound waves propagate at an approximately half of the speed of light. The radius of the sound horizon is 200,000 light-years in ΛCDM and 6 million light-years in Milne?
The horizon radius is 10X too large in Milne?
Actually, the formula
S = = c / a₀ * (ln(t) - ln(t₀))
tells us that the horizon radius is infinite, if we set t₀ to zero. The sound wave can propagate an infinite distance if the universe starts from a point.
What if we claim that the universe starts at, say, t = 10 million years? Then the sound horizon would have an appropriate size – but nucleosynthesis will not work, since the temperature in the synthesis must be around 10⁹ K.
An alternative Milne-like model: a newtonian explosion
Above we defined the Milne model as something which is like FLRW, but has a constant expansion rate. The spatial metric stretches with time.
Another way to make a Milne-like model is to consider a "newtonian" explosion of matter with a positive mass, balanced by strange matter with a negative mass. The explosion cloud keeps growing at a constant rate.
Then the CMB radiation, which we are observing now, has traveled in Minkowski space for ~ 13 billion years. The angular spectrum of the CMB map reveals features of a size ~ 1/60 radians, or 220 million light years. What could these features be? Can they be baryonic oscillations of the plasma?
The last scattering of CMB happened at a time t = 12 million years. The sound horizon cannot have a size of 220 million light years. Is there any way to save this model?
Milne model for times > 380,000 years only?
Most of the mass of the universe in the early stages was radiation, i.e., massless particles. Does the following make sense:
There are particles in dark matter, such that, their interaction, on the average, cancels the effect of the gravity of dark matter on the expansion of the universe.
Those particles might have a negative gravity charge, or an attractive force which produces the negative pressure required to cancel the effect of the gravity of dark matter.
Then the radiation-dominated early stages of the universe would happen just like in ΛCDM, but the dark matter dominated latter stages would happen just like in the Milne model.
But people believe that the radiation domination ended when the universe was just 47,000 years old, or a redshift of 3,600. To reproduce the baryon acoustic oscillations, we need radiation domination to last at least 380,000 years, until the time when the CMB was emitted. Then the universe will develop along ΛCDM long enough.
Let us check how do we know the radiation content of the universe. How much primordial gravitational radiation exists? What is the energy of neutrinos?
The relevant parameter is the "decoupling" time, after which only a small amount of energy flows from the hot plasma to a type of radiation.
As the universe expands, that type of radiation will lose energy and becomes insignificant quite soon.
The estimated temperature T of the neutrino background is now 1.95 K and graviton background < 0.9 K. Since the energy density of a type of radiation is ~ T⁴, then most of the radiation energy content is in photons, whose temperature is 2.725 K.
There might be radiation in dark matter, too. Then the radiation-dominated era could last longer than 47,000 years.
Dark matter particles which pull each other create the negative pressure required for dark energy?
The particles might be massless. Then they would not get caught into gravity wells, and could be distributed quite evenly across the observable universe.
The particles attract each other. They produce enough negative pressure to cancel the effect of dark matter on the expansion of the universe.
Is this plausible?
Let us consider an electron-positron pair. Their common electric field produces a negative pressure. The field acts like the hypothetical "dark energy".
If we imagine that the spatial volume of the universe is finite, then in order to draw the field lines that obey Gauss's law, we must have an equal amount of positive and negative charge.
If the universe is spatially infinite, then there is no need to balance positive and negative charges, since infinity can act as the "other end" for the lines of force.
Would these positive and negative charges clump together and the negative pressure would go away? This has happened with electric charges in our universe.
Dark matter particles with a negative gravity charge?
The energy of the gravity field is negative. Suppose that we have field lines going from a positive gravity charge A to a negative gravity charge B. If we move A and B closer to each other, the field volume shrinks: we conclude that the field energy rises, and there is a repulsive force between A and B.
Negative gravity charges attract each other because the field becomes stronger when we clump them together.
Is there any reason why negative gravity charges should cancel out positive gravity charges during the matter-dominated era of the universe, but not during the radiation-dominated era?
Determining the Hubble constant if the late stages of the universe are Milne
The user "gabe" in the link above (2023) gives the formula as:
where DA is the "angular distance" of the light in the CMB. The sound horizon size rs which we see in the CMB map:
A Milne model at a late stage of the universe means that the Hubble "constant" H(z) at earlier stages is surprisingly small. In a pure ΛCDM model, the Hubble constant would grow faster when we go back in time.
How does a late Milne model affect DA? It is larger. Also rs is larger. Who wins?
The Hubble constant in a matter-dominated universe
Marco Raveri (2023) gives us useful formulae for determining the age of the universe in various cases:
where in the present universe, the following must hold:
to make the spatial metric flat currently. The various Ω are the energy densities of radiation, matter, and dark energy, relative to the critical density right now.
The Hubble value H is defined as
H = (da / dt) / a.
Let us calculate the history of the universe, assuming that
Ωm = 1
today. Then
da / dt = a H(a) = H₀ / sqrt(a).
Let a = 1 now.
Last scattering of CMB. The redshift z is 1,100, which means that a(t) at that time was 1 / 1,100'th of the current a. The speed of the expansion, da / dt, was 33X the current one.
James Webb's largest redshifts z = 15. The size of the currently observable universe was 1 / 15'th of the current one. The speed of the expansion was 4X the current one.
Let us assume that the universe has been almost Milne since the redshift z was something like 100. Then James Webb will see the density of galaxies much larger at z = 15 than in the ΛCDM model.
But the sound horizon size at z = 1,100 must be the same as in ΛCDM, in order to explain the galazy density in the Sloan Digital Sky Survey (SDSS).
To make the universe Milne at z < 100, there has to be some process which slows down the expansion rate surprisingly much around z = 100, and after that keeps the expansion rate unchanged. What could it be?
What about a model in which all matter is counterbalanced by negative mass charges? Then the current spatial metric cannot be flat – but it does look flat.
Let us analyze using the Friedmann equations.
We assume that k = 0 and the matter density ρ and the cosmological constant Λ cancel each other out.
What kind of a radiation density would produce the age 380,000 years at the last CMB scattering?
The radiation mass density is
ρr = C / a⁴,
where C is a constant. Then
da / dt = sqrt(8/3 π G C) * 1 / a.
Observation. With a suitable pressure p, we can tune the energy density ρ(a) of the universe as a function of a as we like. In that way we can make da / dt anything we like, and a(t) as we like! We can create the pressure by assuming forces between dark matter particles.
Thus, we do not need the cosmological constant, or "dark energy" at all. Dark matter and suitable force fields will do the job.
Conclusions
The universe looks somewhat Milne-like at present. The expansion rate is constant, or may be somewhat accelerating.
Also, the James Webb photographs with "too many" galaxies reveal that either the universe at z = 15 is surprisingly old, or we do not see the angular magnification of the ΛCDM. Both are Milne features.
By assuming dark matter particles with suitable force fields between them, we can tune the pressure of the universe in the way that any scale factor function a(t) is created. We do not need to assume a cosmological constant or "dark energy".
The concept of "dark energy", or a cosmological constant Λ, is very speculative. In everyday physics we do not encounter anything which looks like "dark energy". On the other hand, we can easily imagine particles whose mutual attraction creates a negative pressure.
Certain authors have claimed that a variable dark energy negative pressure breaks the dominant energy condition in the universe. But it is not so: as the universe expands, it does work against the attraction between hypothetical dark matter particles. The energy does not appear from nothing.
The predictive power of the dark matter pressure hypothesis is zero: it can explain everything! In a future blog post we will investigate this.
If we assume that the first 380,000 years happened much like in ΛCDM, then there has to be an extra positive pressure after the last scattering of the CMB. The positive pressure quickly slows down the expansion to the current speed, in the year 2024. After that, an extra negative pressure makes the universe roughly Milne. Is there any reason why this would be a reasonable scenario with hypothetical dark matter particles?
Let us run time backwards. A negative pressure from an unknown attractive force within dark matter initially keeps the collapse of the universe happening in the Milne way. When there are some ~10 million years left until the singularity, a positive pressure (= a repulsive force within dark matter) greatly speeds up the collapse. The extra pressure does not grow appreciably as the universe contracts. The effect of the matter density and the radiation density take over during the last 380,000 years of the collapsing universe.
We need to assume two unexplained pressures. This is very ugly. Maybe we have dark matter particles which like a certain fixed distance between them, and resist both the expansion and the contraction from that equilibrium state? That is very ad hoc.
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