The book Galaxy Formation and Evolution by Mo, van den Bosch, and White (2010) is available as a pdf file in the link.
The chapters about Cosmological Perturbations handle the fate of density variations in the early universe.
The mass excess at certain locations in the early universe: the Schwarzschild radius
The current density of baryonic matter in the observable universe is
~ 3 * 10⁻²⁸ kg/m³.
A large galaxy cluster is 10²⁴ meters in diameter, which makes a baryonic mass of
~ 3 * 10⁴⁴ kg.
Let us assume that the "excess" mass is 1 / 100,000 of that, or
M = 3 * 10³⁹ kg.
The Schwarzschild radius of such a mass is
R = 2 G M / c²
= 1.5 * 10⁻²⁷ * M m/kg
= 5 * 10¹² m.
We see that if the scale factor of the universe is
a = 10⁻¹¹,
then the excess mass is inside its Schwarzschild radius. (The factor a = 1 is the present universe.)
The excess mass should form a black hole in the early universe. Why would a black hole expand like ΛCDM claims?
Playing the Big Bang in reverse: density variations form black holes
The ΛCDM model claims that the metric of the early universe is very smooth, spatially uniformly expanding metric. If we have two observers at the edge of excess density volume in the early universe, the observers can communicate much like two observers in Minkowski space.
But is that really possible if the dense volume is inside its own Schwarzschild radius, and is a black hole?
If we assume that gravity is time-symmetric, then we can play the Big Bang expansion in reverse. Any (radio) communication between the two observers can happen in reverse, too.
One of the observers is inside the horizon of a black hole, the other is outside it. We see that the observer inside cannot send a signal to the outside.
This is in contradiction with the claim that the observers can communicate without problems.
A small density perturbation in a middle-aged universe amounts to a huge perturbation in the early universe. This is the well-known problem of the instability of an expanding universe.
Note that the Milne model suffers from the instability just like ΛCDM.
A possible solution: the universe does not begin from a singularity, but is relatively "large" at birth. The diameter of the current observable universe might have been much larger than 5 * 10¹² meters at birth. We do not like singularities in this blog. This could be a step forward, actually.
Does the literature address the instability problem in the early universe?
The increasing energy of the inflaton field breaks the Einstein-Hilbert action?
The basic idea in cosmic inflation is that a scalar field somehow obtains random fluctuations in energy density, and these fluctuations are "blown" spatially very large in the expansion of the universe.
Let us have a history H of the universe. Let us make a variation H' where we keep the metric as is, and thus, the action integral of the Ricci scalar R does not change.
wave
-------______--------_____------- tense string
/ / /
\ \ \ springs
The lagrangian describes an oscillating tense string which also is attached to springs, which create the potential V(φ).
In the inflation hypothesis, φ varies from place to place, and the potential energy (or kinetic energy) is greatly increased as the universe expands.
Let us have a "disturbance" W of the field.
Suppose that the disturbance W is not moving substantially. It is the springs in the diagram which make the string to oscillate up and down.
The expansion of the universe adds more length to the oscillating area, and more energy.
Let us have a history H where the oscillating area expands. Energy is produced from "nothing".
Let us make a history H' where we keep the metric the same. The time varies within an interval Δt.
We make a Noether time variation which "measures" the energy of the system at the start and the end of the time interval Δt. The process is like we would have a harmonic oscillator whose mass is zero, but the energy non-zero, created from empty space at some time t during the time interval Δt.
The Noether time variation changes the value of the action close to the end of Δt, because the oscillator there has energy, but does not change the action close to the start of Δt, because the oscillator does not exist there. We conclude that H' has a changed value of the action. H was not a stationary point of the action. H is not an allowed history.
Suppose then that the disturbance W is moving. A Noether time variation can measure the energy of W at the start of Δt, and at the end. We end up in the conclusion that H is not an allowed history.
We showed that a scalar inflaton field is not compatible with the Einstein-Hilbert action.
Note that if there is no disturbance W, and the field φ simply has a constant potential V, then our counterexamples do not work. It may be that dark energy is compatible with the Einstein-Hilbert action.
*** WORK IN PROGRESS ***
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