slide 1: Citation: Knickelbein MB. The Effects of Inflation within Baryonic Matter. J Phys Astron. 202084:199.
© 2020 Trade Science Inc. 1
The Effects of Inflation within Baryonic Matter
Mark B. Knickelbein
Argonne National Laboratory Ret. 9700 S. Cass Ave. Argonne IL 60439 USA
Corresponding author: Mark B. Knickelbein Argonne National Laboratory 9700 S. Cass Ave. Argonne IL 60439 USA Tel:
+17088486536 EMail: knickelbeingmail.com
Received: October 15 2020 Accepted: November 16 2020 Published: November 23 2020
Abstract
In this contribution the notion that inflation acts on matter only at cosmological length scales is challenged. The generalization of expansion to
microscopic length scales put forth here contrast with the currently accept notion that expansion effects only large massive celestial structures
e.g. galaxies and galaxy clusters over vast regions of space. The effects of inflation operating on microscopic scales within both isolated atoms
and condensed matter are examined. The growth of isolated hydrogen atoms due to inflation is examined using a nonrelativistic quantum
mechanical model. The model predicts that with time the unperturbed atom is put into a superposition state possessing an energy greater than
that of the ground state. The evolving superposition state is predicted to radiatively relax to the ground state within 10
5
s after it is formed with
a distribution of radiofrequency emission peaking at ∼275 Hz. Extension of this conjecture to expansion within stellar matter is considered using
a thermodynamic analysis. It is predicted that expansion within the Sun produces power amounting to 3 to its total luminosity. The results
presented here suggest that expansion on the microscopic scale may have important consequences in astrophysics and cosmology as well as in
theoretical atomic and particle physics where length is assumed to be a timeindependent variable. In particular understanding the effects of
expansion on the properties and behavior of fundamental particles may require modifications of some aspects of the Standard Model.
Keywords: Cosmology Hubble constant Inflation Interstellar hydrogen Atomic physics Quantum mechanics Stellar luminosity
Introduction
Inflation is a centerpiece of our current understanding of the structure and time evolution of the Universe. The notion of cosmological
inflation initially proposed in the wake of astronomical observations by Hubble and coworkers 1 was invoked to account for the
correlation of spectral red shifts of galactic light and the apparent distance of those galaxies from the Earth. Since that time inflation has
been a central concept in the development of cosmology—the expansion of the Universe since shortly after the Big Bang 23. In the past
few decades much progress has been made in the interpretation of inflation within the Standard Cosmological Model using General
Journal of Physics Astronomy
ReviewVol 8 Iss 4
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Relativity as a theoretical framework 48. The basic notion of exponential spatial inflation has stood the test of time and is now central to
our understanding of the Big Bang and its aftermath in the evolution of the Universe.
The exponential expansion observed in the Universe is most commonly described by the Hubble–Lemaître law commonly known as the
Hubble Law as parametrized by the Hubble Constant H
0
6. The value of the Hubble constant H
0
is being constantly refined as
experimental techniques and analysis methods improve and there is currently some discrepancy between values determined using different
approaches 9. However these discrepancies are of the order of only 10 percent and since the precise value of H
0
is not important to the
general conclusions gleaned from the following analyses we shall use the most recently accepted value 6:
H
0
68 ± 2 km s
1
Mpc
1
In discussing inflation at microscopic length scales it is more convenient to use the rate constant form of the Hubble constant containing
no reference to a particular length:
H
0
6.95 × 10
2
Gyr
1
H
0
2.20 × 10
18
s
1
Any length l of space is then predicted to expand exponentially with time as
Note that when H
0
is expressed in this way H
0
1
14.4 × 10
9
years—a value not far from the estimated age of the Universe since the Big
Bang: 13.8 × 10
9
years 3.
If we accept that the ―fabric‖ of space is continuous at every length scale then it becomes interesting to consider the effects of inflation at
the microscopic level in particular how inflation effects the internal structure of isolated atoms and other baryonic matter in the Universe.
Hydrogen is estimated to make up 90 of the baryonic matter of the Universe 410. The majority of hydrogen is contained in the
interstellarand intergalactic media some as neutral hydrogen atoms some as ionized hydrogen—bare protons p and electrons e in low
density plasmas 11. In the intergalactic medium the density of hydrogen is estimated to be of the order of one atom or proton per cubic
meter. At this density the mean time between collisions HH Hp HHe pe etc. is 10
7
10
9
years varying with the density and
temperature of the medium and the collision partner. A gas kinetics approach that can be used to make simple meantimebetween
collision estimates is outlined in the Appendix. Neutral hydrogen atoms in the intergalactic environment are thus relatively unperturbed on
the timescale of the estimated age of the universe 1.4 × 10
10
yr 3.
Because the effects of inflation on microscopic systems have not to the author’s knowledge been considered previously we must
improvise using the theoretical tools available. In the following analysis we consider the effects of inflation on the internal space within
the isolated hydrogen atom using the methods of nonrelativistic quantum mechanics. We then consider the effects of inflation within the
dense matter comprising stellar interiors using a classical thermodynamical model.
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Expansion within the hydrogen atom
The notion that the empty space within microscopic baryonic matter expands with time is conceptual terra incognita. So we can only
proceed with a theoretical analysis of this phenomenon using the tools we have available. In the case of the isolated hydrogen atom we
will use wellestablished methods of quantum mechanics.
The hydrogen atom in its ground state has a volume of ∼10
31
m
3
using the Bohr radius a
0
0.0529 nm as an approximate but well
defined measure of its size. In contrast the volume of hydrogen’s nucleus the proton is ∼10
45
m
3
while the electron a lepton has in
comparison no or negligible volume. Thus the hydrogen atom is almost entirely made up of empty space. The expansion of the internal
volume of the hydrogen atom with time will be treated assuming that the Hubble expansion law is valid at atomic length scales. With the
assumption of exponential spatial expansion as given in Equation 1 the radius of the hydrogen atom R initially at R
0
will then increase
exponentially with time like any other distance of space:
Because the hydrogen atom in its ground 1s state has no welldefined radius in the classical sense we can simply assign R
0
to the value
of the Bohr radius a
0
or to the quantum mechanical expectation value of the protontoelectron distance r
1s
3a
0
/2. The assignment of
any precise value for its radius is not critical for this discussion so in the following analysis we shall simply equate R
0
to a
0
. Equation 2
predicts that in the absence of external perturbations the hydrogen atom’s radius however it is defined would increase by about 7 and
its volume by about 23 in 10
9
years 1 Gyr. This growth conjecture based on Equation 2 predicts that unperturbed hydrogen atoms will
simply grow exponentially without bound however this notion is not consistent with the true nature of atoms which are properly described
by quantum mechanics. In particular the electron radial probability density of the hydrogen atom in its ground and excited states are well
understood within quantum mechanical theory defined by the appropriate wave functions and are not subject to arbitrary ―adjustment‖.
Here we use the nonrelativistic formulation.
To put intraatom expansion of hydrogen on a firmer theoretical footing we first cast the problem in classical form in which we take a
0
to
be the initial radius of the normal groundstate atom. The energy of the ground state E
0
is given by
where E
0
13.6 eV 12. Increasing the internal volume of the atom amounts to increasing the mean protontoelectron distance r from its
most probable value groundstate value a
0
to an incrementally larger value a´. This radial expansion is accompanied by an incremental
increase of potential energy ΔE
V
given as the difference in Coulombic attraction:
Using the tools of timeindependent perturbation theory 1314 the corresponding energy perturbation term in the Hamiltonian Ĥ´
contains only a´ and a
0
with no differential quantum mechanical operators and is thus identical in form to the classical expression
Equation 4
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Using the exact ground state wave function for the hydrogen atom ψ1s we then calculate the firstorder correction
E
V
ψ
1s
 Ĥ´ ψ
1s
to the ground state energy E
0
. Using the perturbation Hamiltonian given by Equation 5 the first order energy correction E
V
is given as
The perturbation theory result given by Equation 6 is found to be identical to the classical result given by Equation 4 except reversed in
sign reflecting the addition of energy upon increasing the atom’s radius E
V
is a positive quantity. Thus the effect of expansion is to add
energy to the atom due to the increased potential energy of the expanded system. The timeevolution of the added energy E
V
t is found by
substituting the growth expression
for a´ in Equation 6:
Equation 7 predicts that as t→∞ E
V
approaches a
0
1
giving the total energy E
tot
E
0
+ E
V
for the atom:
The asymptotic value for E
tot
given by Equation 8b is +13.6 eV clearly at odds with the expected total energy for infinite protonelectron
separation 0 eV. The discrepancy can be traced to the neglect of the electron’s kinetic energy T in the analysis: the total energy E
tot
must
approach zero as the potential energy E
V
approaches its maximum at infinite separation. The quantum mechanical virial theorem as
applied to the hydrogen atom predicts that the expectation values of the potential energy V and kinetic energy T are related to the
total energy E
tot
as follows 1314:
Thus
Following this result we reformulate an expression for E
tot
that accounts for the electron kinetic energy adapting the virial theorem to
account for the changes in T and V as the atom expands:
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In Equations 10a and 10b the first term is the total groundstate energy of the unexpanded atom the second term is the change in potential
energy E
V
t and the third term is the change in kinetic energy E
T
t E
V
t/2 obtained using the virial theorem adaptation. With this
modification E
tot
exhibits the expected asymptotic behavior as t→∞:
The time evolution of E
tot
as predicted by Equation 10c is shown in FIG. 1. Also shown in FIG. 1 are the computed expectation values
r
1s
r
2s
r
3s
and r
4s
along with the experimental energies for those states. Note that the E
tot
curve passes smoothly through the
wellestablished energy eigenvalues corresponding to the r
ns
.
FIG. 1. Total energy of the hydrogen atom as a function of time as predicted by Equation 9. The top axis shows the
corresponding size of the atom as a function of r/a
0
. The solid circles are the r
ns
and E
ns
E
0
n
2
values computed for n 14.
The quantum mechanical treatment applied here gives us a simple picture of the hydrogen atom that becomes incrementally inflated with
time but does not address its ultimate fate. This analysis predicts unlimited exponential expansion but that is physically unrealistic. In our
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model of isotropic spherical expansion the mean protontoelectron distance r would seem to increase without bound. However the
excited states with n1 would eventually interrupt the expansion by providing a means to return to the ground state via dipoleallowed
transitions. In particular when r in our expanding atom approaches that of the 2 s state r
2s
6a
0
the atom would be expected to
return to the ground state by emission of an ultraviolet photon via the Lymanα transition λ 121.6 nm hυ 10.2 eV 12. The time t
2s
required for the atom to increase in size from the ground 1 s state to the 2 s state as measured by r
2s
is calculated to be 19.9 Gyr—
greater than the estimated age of the Universe. But how do we describe these atoms between 1 s and 2 s To address this we turn again to
the available tools: the quantum mechanics and spectroscopy of timedependent systems.
A hydrogen atom that has become expanded in volume even incrementally is clearly no longer in its ground state but rather must be
described quantum mechanically as a timedependent superposition state Ψt 1314. The contribution of the excited n1 states to this
state increases continuously as the dimensions of the atom increases with time. Formally we express the electronic structure of this
superposition state as of a linear combination of the ground and excited states of hydrogen:
∑
where ψ
ns
are the ground n 1 and excited n ≥ 2 sstate wave functions. Note that the expansion coefficients c
n
are in general complex
and timedependent. Atomic hydrogen 2 s1s superposition states have in fact been produced in the laboratory in ultrafast laser excitation
experiments 15. However in these experiments their existence is fleeting so their properties cannot be studied in detail. Because the
expansion of space within the atom is assumed to be isotropic and because there is assumed to be is no mechanism for change of angular
momentum for the isolated atom so as to produce states of nonzero angular momentum: p d f… we consider only the hydrogenic s
states in the linear combination Equation 12 thus preserving the spherical symmetry of the atom. In this simple picture the effect of
spatial expansion is to contribute excited s states into overall wave function Ψt. Because this superposition state contains contributions of
excited s states there must exist a nonzero probability for dipoleallowed transitions to the ground state that increases with the fraction of
excitedstate character in the inflated atom that is as the atom increases in size. We therefore consider the dynamical properties of these
superposition states Ψt. Because nothing is known about the dynamical e.g. radioactive properties of superposition states in atomic
hydrogen we will first simplify the problem by initially only considering the 1 s and 2 s states but with the implicit understanding that all
ns states actually contribute to the superposition state as given by Equation 12. In this simplified model we will denote the 2s1s
superposition state as:
Whereas the complex coefficients c
n
t give the contribution of each basis wave function ψ
ns
in the overall wave function Ψt it is the
c
n

2
c
n
c
n
terms that provide the connection to the physical observables such as electron probability density and radioactive properties
1314. The c
n
t then must satisfy the following criteria:
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1. Following our physical description of the growth of the atom see above c
2
t
2
must increase exponentially from 0→1 as t varies from
0→t
2s
where t
2s
6.28 × 10
17
s 19.9 Gyr. That is at t
2s
6.28 × 10
17
s the atom reaches the dimensions of the 2s state which is a factor
of 4 larger than the ground state r
2s
/r
1s
4
2. c
1
t
2
must decrease with time as c
1
t varies from 1 at t 0 to 0 at t t
2s
3. Completeness must be satisfied: c
1
t
2
 + c
2
t
2
 1
Using criterion 1 it is readily shown that
satisfies these requirements between t 0 and t t
2s
. Then by the completeness requirement Criterion 3
c
1
t
2
1c
2
t
2
Now proceeding further into terra incognita we must make some assumptions regarding the radiative properties of the superposition state
in particular how its radiative decay rate varies with the composition of Ψt at any given t. We begin by defining a firstorder time
dependent rate coefficient k
21
t that describes the emission rate of the 2s1s superposition state given by the decay rate of the ―inflated‖
population. Lacking any available theoretical model to guide us in this problem we will simply assume that k
21
t varies in proportion to
the square of the fraction of ψ
2s
in Ψt: k
21
t c
2
t
2
Equation 13. The radiative decay rate τ
1
of the pure 2 s state A
21
is 4.70 × 10
8
s
1
for the Lymanα 2s→1s transition 12. Because k
21
t is assumed to vary from 0 for the atom in its 1s ground state at t 0 to A
21
at t
t
2s
when the atom has reached the 2 s state it follows that k
21
tc
2
t
2
A
21
. From Equation 14 it is seen that for small t c
2
t
2
H
0
t/3.
Thus in the small t limit k
21
t ≅ A
21
H
0
t/3
In the following analysis we shall see that the assumption of small t tt
2s
is justified.
The emission spectrum generated by radiative decay of the superposition state Ψt is calculated using a rate equation approach assuming
an initial population N0 of ground state hydrogen atoms at t 0. We redefine the zero of energy to be that of the ground state so that
E0 0
and
The emission spectrum is defined by the radiative decay rate of the evolving population NE as a function of time and energy:
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The time evolution of the emission intensity predicted by Equations 15a15c is shown in FIG. 2. The model predicts that the emission
intensity peaks at ∼4 × 10
4
s more than 12 ordersofmagnitude smaller than t
2s
itself. Thus radiative decay dominates unlimited spatial
growth returning the ensemble of atoms to its original 1s ground by ∼2 × 10
5
s at which point the growth process begins again.
FIG. 2. The emission spectrum predicted for a population of hydrogen atoms born in their normal 1 s ground state at t0. Red
curve: Emission spectrum calculated assuming a superposition state contribution from 2 s only Blue curve: Spectrum calculated
assuming a superposition state contribution from 2 s 3 s 4 s and 5 s states. Note that the emission frequencies of the inflating
ensemble of atoms are timedependent: the bottom axis shows the time dependence of the emission intensity the top axis shows
their emission frequencies. See Equations 1214 for details of the model.
For the radiative rate analysis thus far we have truncated the superposition series given by Equation 12 and considered only the 2 s excited
state in the derivation of the emission rate expression given by Equation 15a15c. Inclusion of additional higherlying ns states n 3 4
5… in the analysis leads to additional rate coefficients k
n1
t each contributing to the overall emission rate to the ground state in parallel
with k
21
. We define k
tot
as the sum of the individual rates:
k
tot
Σ k
n1
n 2∞
The rate expressions given by Equations 15a15c thus provide a lower bound to k
tot
. Calculation of k
n1
for n 35 using the same analysis
given above for n 2 using the accepted A
n1
emission rates 12 show that these rate coefficients decrease rapidly with increasing n:k
31
is a
factor of 20 smaller than k
21
while k
41
is an orderofmagnitude smaller than k
31
and so on. The emission spectrum calculated for k
tot
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including the n 25 excited states in the analysis is shown in FIG. 2 along with that calculated using only n 2. Comparison of the two
spectra shows only a small shift to lower emission times and lower frequencies when higher n states are included confirming that k
21
is
the dominant term in k
tot
.
The emission frequencies predicted with this model also shown in FIG. 2 lie in lowestenergy portion of the radiofrequency region many
ordersofmagnitude lower than that of the Cosmic Microwave Background CMB and other galactic and extragalactic radiofrequency
emissions that have been detected 16. The predicted emission rate is sufficiently rapid that competition from collisional relaxation would
not be a complicating factor even for hydrogen in dense molecular clouds—including those within our own Galaxy 11. Unfortunately
experimental verification of the predicted spectrum is not possible from Earthbased instruments because the predicted emission
frequencies lie in the radiowaveopaque region of the RF spectrum where radiation does not penetrate the ionosphere 16. Thus space
based probes incorporating the appropriate instrumentation would be required to detect the ultralowfrequency RF emissions predicted in
this analysis.
Expansion within condensed matter
Stars comprise the overwhelming majority 99 of the condensed matter in the known Universe 1. The pressures within normal stars
are enormous from our terrestrial pointofview. The pressures within our Sun for example vary from 10
16
Pa at the center r 0 to
10
11
Pa at r 0.9 R⊙ finally approaching zero at the surface r R⊙ 17 the atmospheric pressure on Earth at sealevel is by contrast
only ∼10
5
Pa. However even at a normal star’s center it is easily calculated that the constituent matter is largely ―empty space.‖ If we
accept the idea that this empty space within the interior of the Sun and other stars expands at a linear rate H
0
we are compelled to consider
the energetic implications. In particular we must devise a means to calculate the energetic consequences of space being continuously
―created‖ within the highpressure environment of a star due to inflation. While the creation of volume within condensed matter due to
cosmological expansion has no classical analog we nevertheless adapt a thermodynamic model here to describe this phenomenon. The
creation of an increment of space within stellar matter due to expansion will produce a nonequilibrium expanded state which is assumed
to relax instantaneously. Given the large collision rate the expansionrelaxation cycle will be so rapid that it would seem that at first glance
that there should be no observable effect. However this continuouslyoccurring process must be considered as a source of internal power
to the star due to PV work adding to the power that is produced via fusion processes in the stellar core. To calculate this additional power
source we will proceed with a thermodynamic approach.
The generalized integrated form of the Hubble expression describing expansion of any given initial length l
0
initially at t 0 is
Equation 1 so that the expansion of any arbitrary volume V
0
l
0
3
is simply
The increment of PV work dW
PV
generated within volume dV at a pressure Pr and at a distance r from the center of the star then is then
with the corresponding increment of power dP
PV
given by
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where Equation 18b was obtained using the expression for Vt given in Equation 16. Because it is assumed that relaxation of the non
equilibrium expanded state of stellar matter is instantaneous we let t 0 in Equations 18 and set the initial volume V
0
to the spherical
volume element at radius r:
V
0
r 4πr
2
dr
Then integrating the resulting expression over r we obtain an expression for the total power P
PV
:
∫
Our Sun a typical star whose properties are as wellknown as any has a pressure profile Pr predicted by the Standard Solar Model 17.
Using the tabulated values of Pr vs. r obtained from this model we evaluate P
PV
by simple numerical integration of the integral on the
rhs of Equation 19 and obtain the final result: P
PV
1.3 × 10
35
W. This amounts to 3.4 of the total luminosity of the Sun L⊙ ≅ 3.8 ×
10
26
W 18. Thus expansion is predicted to contribute to the power output of the Sun in a small but nonnegligible way. Finally it is
noted that this same analysis would not apply to neutron stars as they do not contain any significant ―empty space‖ within them for
expansion to occur—they are thought to be composed of voidless ―solid‖ baryonic matter sometimes referred to as ―neutronium‖.
Summary
In this paper the notion that the microscopic internal space within baryonic matter expands with time has been put forth. The author
stresses that the analyses in this paper are by no means unique: they are simply besteffort calculations based on what we know as applied
to what we do not. For isolated matter such as intergalactic atomic hydrogen the internal expansion conjecture might be dismissed off
hand by invoking the occurrence of a relaxation mechanism continuously acting to maintain the atom in its ground uninflated state. With
that assumption space expands over time but atoms fly through it unaffected forever displaying their original ground state internal
dimensions and energies. However such continuous relaxation would require a medium to serve as a ―heat bath‖ and in the absence of
such a medium there is no mechanism for mediated relaxation—the only relaxation mechanisms must be intrinsic to the atom as space
expands within it. It was once thought that such a medium existed necessary to support the propagation of electromagnetic waves through
space. However Michelson and Morley demonstrated the absence of such an ―ether‖ through their landmark interferometry studies. The
―stretching‖ of the highenergy photons present at the dawn of transparency in the very early Universe to the CMB that we observe now is
an example of inflation acting to increase the wavelengths of photons thus reducing their momenta and energies over the course of time
without any medium or other intervening thirdbody mechanisms. Can the same notion of dimensional stretching operate in isolated
baryonic matter
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Within condensed matter a heat bath is ―built in‖ in the form rapid collisions between particles enabling the relaxation to equilibrium of
any nonequilibrium state produced by internal expansion as would occur in any fluid. The net result is heat production due to the PV
work done on the system. As applied to the Sun the resulting heat production is predicted to amount to a few percent of its total
luminosity. While this is a modest fraction of the total output of the Sun perhaps power production due to expansion within darker lower
temperature stars might contribute a larger fraction to their luminosities.
Appendix
In order to assess whether or not an atom in an excited state undergoes collisional relaxation to the ground state in a given time we need to
know the probability that the atom undergoes a deactivating collision with another atom or molecule proton electron etc in that time.
Assuming that the gases in interstellar and intergalactic media are at local thermodynamic equilibrium we can calculate their collision
rate constant k
c
and hence meantimebetweencollisions using the wellestablished kinetic theory of gases 19. In its simplest form k
c
is
given by the product of the mean relative velocity between any two atoms in the ensemble v
r
and the collision cross section σ
c
:
The collision cross section σ
c
is calculated as πr
c
2
where r
c
is the effective collision radius of the colliding pair. For HH collisions for
example we would reasonably estimate the collision radius as the radius of the H atom0.0529 nm the Bohr radius a
0
in which case σ
c
8.8 × 10
21
m
2
.
The mean relative velocity v
r
of two colliding atoms within an ensemble at temperature T is given 19 by:
⟨
⟩
where µ is the reduced mass of the colliding pair. For the case of HH collisions µ 8.3 × 10
28
kg. For ease of calculation Equation A2
can be then expressed as σ
c
4.2 × 10
4
T
1/2
ms
1
for HH collisions. The collision rate R
c
experienced by a single H atom is given by
where n is the number density of collision partners here taken to be atomic hydrogen.
Using this model it is calculated for example that in a gaseous medium in which the density of hydrogen is 1 m
3
and the temperature is
100 K the collision rate experienced by one H atom R
c
3.7 × 10
17
s
1
1.1 collision per Gyr and the mean time between collision R
c
1
2.7 × 10
16
s 0.9 Gyr
1
. One can easily extrapolate the calculation of R
c
to other values of n and T by noting that R
c
is proportional to n
and to √T. Likewise σ
c
for collision partners other than hydrogen atoms can be estimated using standard bimolecular collision models 19.
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