# An energy-based natural selection model

Noam Abadi<EMAIL_ADDRESS>Guillermo Abramson<EMAIL_ADDRESS>Centro Atómico Bariloche (CNEA), R8402AGP Bariloche, Argentina  Instituto
Balseiro, Universidad Nacional de Cuyo, Argentina  Consejo Nacional de
Investigaciones Científicas y Técnicas, Argentina

###### Abstract

Genetic information and environmental factors determine the path of an
individuals life and therefore, the evolution of its entire species. We have
succeeded in proposing and studying a model that captures this idea. In our
model, a renewable resource extended throughout the environment provides the
energy necessary to sustain life, including movement and reproduction. Since
the resource doesn’t regrow immediately, it generates competition between
individuals and therefore provides a natural selection pressure from which
evolution of the genetic traits is observed. As a result of this, several
phenomena characteristic of living systems emerge from this model without
having to introduce them explicitly. These include speciation and punctuated
equilibrium, competitive exclusion, and altruistic behaviour from selfish
rules.

###### keywords:

biological evolution , competition , natural selection , genetic algorithms

††journal: Physica A

##  1 Introduction

Darwin’s theory of evolution [1] was a huge step towards answering questions
regarding the origins of life and providing a unifying framework for its
diversity as observed on Earth. However, the theory is so broad in its reach
that many aspects of it are still under study. For example, it has become
increasingly accepted that the evolution of the biosphere needs an integrated
description of both the biological agents as well as their resources [2].
Indeed, the main agents and processes responsible for the evolution, namely
speciation and extinction, are the result of interactions, and interactions
are often mediated by shared resources.

The main area where modern understanding transcends Darwin’s is genetics. We
understand many details of the inheritance of traits, from the molecular to
the physiological levels. Yet, the relation between genetically inherited
traits and the probability of producing offspring in a certain environment is
not completely clear. Similarly, it is not obvious what general features of
extinction, survival, and eventually, speciation should arise from the complex
interaction of the many mechanisms involved. Many of these problems have been
studied in recent decades from a theoretical point of view, with a variety of
tools rooted in the theory of complex systems [3, 4, 5, 6].

Moreover, living systems are chemical, and their dynamics is driven by free
energy. At the core of life, there seems to be just a chemical metabolism,
which nowadays is a complex network of reaction pathways, involving both
equilibrium and out-of-equilibrium processes that drive the flow of free
energy through the system. Indeed, there is no indication that individuality
or replication played a role in early chemical (pre-biotic) organization [7,
8]. Even if they perhaps do not _define_ life, individuality, replication, and
selection arose at some moment in the evolution of living systems as
persistent and universal distinctions. In this work, we propose a model in
which individual agents have traits, determined by genes, that determine the
balance between energy input and output. We study how a population of such
creatures evolves, in a scenario of competition for a spatially extended
common resource that provides the necessary energy. The capacity to move is
one such trait, and it allows creatures to change their position when they
exhaust their local resource, albeit at a cost in energy. Likewise, body mass
is energetically expensive to keep alive and also to carry around but
guarantees a reserve in case of resource scarcity. Since both traits have
benefits and drawbacks, we expect them to have optimal values, which the
system should achieve in a stationary state, in a context of competition
mediated by the common resource.

The model is similar to latent energy environment (LEE) models [9, 10], but
with two key differences: first, the resource is entirely equivalent to a
creatures energy instead of components that need to be combined with others to
produce it. This allows the second distinction, which is that individuals have
no neural network to act as a brain and allow them to learn throughout their
lives. The advantage of this is that the model can address questions regarding
primordial life forms.

As opposed to artificial selection, which evolves towards the optimization of
a certain function [11], natural selection evolves under a certain constraint,
which we propose is the limitation of energy. Therefore, the model does not
prescribe any landscape of fitness, nor any function to be optimized. Traits
are inherited from one generation to the next with small, random mutations,
that may change their performance and the corresponding energy balance. This
makes the population undergo a natural selection process as the agents
interact through the common resource. Selection is not modeled, but an
emergent mechanism that provides a certain organization to the ecosystem.

We show that phenomena commonly associated with natural selection appear to be
intrinsically tied to it in our model. The first one is speciation, in the
form of two possible sets of traits in a stationary state, which is achieved
by mechanisms akin to those of punctuated equilibrium. The second one is that
the main behavior throughout the population is much more altruistic than what
is expected from individual rules, as suggested by the selfish gene theory,
both in the amount of resource consumed and life expectancy. We also provide a
mean-field description of the resource dynamics, which shows important
consequences of its spatial extension and limited growth.

##  2 The hydra model

We will proceed to formulate our model111We call it the “hydra model” because
of superficial similarity with the freshwater genus of the same name: tumbling
motion, asexual reproduction by budding and potential immortality. Details of
the model, either particular parameter values or results, do not pretend to
correspond with actual specific features of the life history of _Hydra sp._
based on the following key ingredients of an evolutionary system as identified
by Kaneko [12]: 1) Genotype, phenotype, and environment; 2) Fitness depends on
the phenotype and the environment; 3) Only genes are transferred to the
offspring; 4) Mutation of genes occurs randomly without any specific
direction. Moreover, in the formulation that we describe below, fitness is not
a prescribed function but an emergent feature of the dynamics, as will be
discussed further on.

Consider a grid of size $L\times L$, and a resource $f$ extended over it. Let
us suppose that, in the absence of consumers, its dynamics obeys a locally
logistic equation [13]:

$\frac{d\,f(\mathbf{x})}{dt}=rf(\mathbf{x})\left(1-\frac{f(\mathbf{x})}{K}\right).$
(1)

A population of individuals lives on the grid and moves around it, feeding off
the resource to obtain the energy $E$ needed to stay alive and storing it in
their bodies. For each creature this energy evolves according to intake and
consumption

$\frac{dE(t)}{dt}=\varphi(t)-C(t),$ (2)

where the feeding function $\varphi$ is the amount of resource that the
creature gets from the cell on which it is located, and the consumption $C$ is
the amount of energy it uses to fulfill its bodily needs. Equation (2) states
that energy cannot be created inside a creature’s body, and therefore has
little meaning without modelling $\varphi$ and $C$. To this end, we define a
genotype ${\bf G}=\\{M,P,s\\}$, which consists of the adult body size, $M$,
the adult probability of moving to a nearest neighbour cell, $P$, and the
growth rate from birth to adulthood, $s$. Upon reproduction, the genotype is
transmitted to the next generation with small mutations, allowing the
population to evolve. The genotype is expressed as a time (age) dependent
phenotype ${\bf F}(t)=\\{m(t),p(t)\\}$, which is the creature’s size $m(t)$
and its probability of moving $p(t)$. These change as a creature ages to
simulate physical growth, according to:

$\displaystyle\frac{dm(t)}{dt}$
$\displaystyle=s\,m(t)\left(1-\frac{m(t)}{M}\right),~{}~{}~{}~{}m(0)=0.05M,$
(3) $\displaystyle p(t)$ $\displaystyle=\frac{m(t)}{M}P,$ (4)

where we have modeled the movement of the creature as proportional to its
mass, corresponding to steps defined by a characteristic length. An exponent
in the relation between $p$ and $m$ can be used to represent other geometries
or movement characteristics.

The phenotype defines $\varphi$ and $C$ in the following way. The feeding
function $\varphi$ is such that a creature will eat enough from the resource
$f$ available at $\mathbf{x}$ to fill its energy storage (of maximum size
$m(t)$, the current mass) or, if the resource is not enough for this, whatever
is available. The available resource was limited so that a small amount of
resource ($0.01K$) always remains in each cell to prevent reaching the
(unstable) equilibrium $f=0$ that is unable to regrow:

$\varphi(t)=\min\big{(}\,m(t)-E(t)\,,\,f(\mathbf{x},t)-0.01K\,\big{)}.$ (5)

The consumption $C$ involves three terms, which represent different bodily
needs. Firstly, we have metabolism, with an allometric dependence on mass that
is found in living creatures across many orders of magnitude of body size
[14]. The second one is kinetic energy, necessary for movement at velocity
$v(p)$. We model this as a stochastic process taking the value $1$ when the
creature moves from one cell to another (with probability $p$) and $0$
otherwise. Finally, we have the cost of growing, i.e. the power a creature
consumes in increasing its mass from birth to adulthood. In mathematical form:

$C(t)=\mu m^{\nicefrac{{3}}{{4}}}+\kappa
mv^{2}(p)+sm\left(1-\frac{m}{M}\right).$ (6)

Coefficients $\mu$ and $\kappa$ in the metabolic and kinetic terms are
proportionality coefficients that may be interpreted as efficiencies; they are
constant throughout the populations evolution and do not play a relevant role
in the model. We have chosen as $\mu=0.1$ and $\kappa=0.5$.

The final mechanisms that need to be defined for the population to undergo
natural selection are death and reproduction. Death happens when a creature
empties its energy storage ($E=0$) or spontaneously with an age-increasing
probability to include the possibility of death by old age. The value of this
probability was set to give an overall expected lifetime of $\lambda=1000$
time steps; we will characterize it thoroughly in Section 3.3.

Reproduction proceeds asexually, simulated phenomenologically as follows.
First, we require an adult individual ($m(t)>0.95M$) that keeps its energy
above a threshold $T_{R}=0.8M$ for $\tau=100$ time steps. After this initial
stage, the body size begins increasing, following a logistic dynamics similar
to the one it followed for its own growth, but with a limit value of $2M$.
This dynamics halts at $m(t)=1.05M$, when the offspring separates from the
parent, producing two distinct individuals. The brood starts its autonomous
existence with mass $0.05M$ and the parent is left with the remaining $M$. The
offspring inherits a genome $\bf G^{\prime}$ which is a mutation of the
parent’s genome $\bf G$, drawn from a random, uniform distribution centered at
$\bf G$ with a maximum mutation of $5\%$ in each gene. We would like to remark
that most of the reproduction parameters were chosen arbitrarily and may not
represent realistic values. The values were chosen so that the relevant
phenomena could be analyzed in a reasonably clear way. The parameters are also
left untouched by mutations since it is difficult to establish how their
effects might be beneficial in one way but detrimental in others. In absence
of optimal values, the system may be susceptible to never reaching a
stationary state.

##  3 Results

The hydra model was studied mainly through numerical simulations. No
additional assumptions were made upon implementation aside from those
described in Section 2. Results were obtained by measuring demographic
characteristics like gene distributions and total population, as well as
resource availability at periodic intervals during the simulation. The
experiments were repeated several times (typically 20) to identify persistent
features in the stochastic realizations. The resource recovery speed $r$ was
varied over sets of realizations to study the evolution under different
conditions. In this section, we present these results and our interpretation
of them.

###  3.1 Population and genetics

Figure 1: Population dynamics. For each value of the resource recovery $r$, 10
runs of the simulation are displayed, to show the existence of two different
stationary states, as evidenced by the formation of two clouds of points.

Figure 1 shows the temporal evolution of the population for different values
of the resource recovery speed, $r$. The simulations begin with $1000$
individuals distributed randomly over both the physical and genetic spaces.
The population dynamics shows three distinct regimes (note the logarithmic
scale in the time axis). The first one is a wipeout of any genes too unfit for
survival, either because of competition with others or because of their own
depletion of local resource and energy reservoir. The second one is a
population explosion where the surviving genes reproduce while the capacity of
the system allows them to do so. The last one is a stationary population value
with no genetic change but constant individual renewal. Both transient regimes
occur very quickly in terms of elapsed generations, which is reminiscent of
the bursts in which evolution is thought to happen in nature [15, 16, 17, 4].

From now on, let us focus on the stationary state unless stated otherwise. A
faster-growing resource means that there is more food available and less need
to move, so the population increases and the adult speed decreases as the
resources recovery speed does. Additionally, both present two branches that
can be interpreted as two possible traits or species that may emerge given a
recovery speed. These features of the stationary regime are shown in Fig. 2.
Bear in mind that the top branch of one plot corresponds to the bottom one of
the other. This means that a larger population must consume less energy for
the resource to be able to sustain it. We will address this inverse
relationship more in detail in Section 4.

Figure 2: Left: Stationary population vs resource recovery rate. Both branches
are linear in $r$. Since the bottom branch is constant in probability of
taking a step, its slope can be calculated with the mean-field equation (19)
obtained in Section 4. Right: Stationary adult speed vs resource recovery
rate. The top branch is constant in $r$, $P=1$ and corresponds to the bottom
branch of the left figure. We found that the bottom branch obeys $P\propto
r^{-1/2}$ and corresponds to the top branch of the left figure.

The adult mass, on the other hand, reaches a stationary value that is
independent of the recovery rate of the resource, within the precision
achieved in the simulations. In a phenomenological linear fit $M=M_{o}+ar$, we
obtained $M_{o}=0.0717\pm 0.0004$ and $a=0.02\pm 0.03$. For values of $r$ in
the simulated range, the error in the constant term is of the same order as
the value of the linear one, making the latter negligible. Meanwhile, the
growth speed $s$ increased monotonously in all the simulations. The specific
form in which this parameter changed was the same for all simulations
regardless of the value of $r$, meaning that a faster growth speed always
increases chances of survival. The effect of this gene seems to be purely in
the amount of time over which a fixed amount of energy, determined by the mass
gene, is expended on growth. Since the behavior of both of these parameters is
untouched by the parameters we varied, we have mostly ignored them in this
analysis.

Figure 3 shows the power spectrum obtained from the population dynamics. It
displays a power-law behavior of the type $f^{-\alpha}$ with $\alpha\approx
1$. This means that population fluctuations are very similar to the ones
obtained in simpler models [4] in which fitness has no physical interpretation
but natural selection takes place nevertheless.

Figure 3: Power spectrum of population fluctuations obtained from population
dynamics in the stationary state.

###  3.2 Altruistic behavior

We have observed the emergence of a rudimentary altruistic behavior from
egoistic rules, in the sense proposed by the “selfish gene” theory [18, 19].
The feeding function modeled by Eq. (5) might suggest that feeding would
always be as large as possible, limited only by the capacity of the creature
or the availability of the resource. However, interactions in the system
manage to provide a setting in which individuals can maintain the minimum
amount of energy needed to reproduce instead of the maximum energy possible.
Figure 4 portrays this effect by giving an estimation of the energy
distribution throughout the population. Instead of being concentrated around
$E/M=1$, it stays just above the reproduction threshold $T_{R}/M=0.8$ (note
the logarithmic scale). This is altruistic both towards other creatures, that
are left with more available resource, and with the resource, by preserving
part of it for its recovery.

Figure 4: Energy distribution measured over a stationary population. Note that
most individuals have energy a little over the reproduction threshold
$T_{R}/M=0.8$. Three values of the recovery are shown, as indicated in the
legend.

###  3.3 Age distribution and death

We will now fulfill our promise to characterize spontaneous death more
thoroughly. First, consider the probability of being dead at age $t$, which we
will denote $P_{D}(t)$. It is the cumulative distribution of the probability
of death at age $t$, $p_{D}(t)=dP_{D}(t)/dt$. The probability of death is also
related to the rate of lethal events, $\rho_{D}(t)$. The latter represents
dangers that may or may not kill an individual at each moment in time. Simply
put, the rate of lethal events is to the probability of death as the outcome
of a single coin toss is to the process of tossing a set of coins until they
fall heads up. The probability of being dead is the proportion of coins that
have stopped being tossed. Since it is a time-dependent Poisson process, we
have:

$\displaystyle p_{D}(t)=$
$\displaystyle\rho_{D}(t)\,e^{-\int_{0}^{t}\rho_{D}(t^{\prime})dt^{\prime}},$
(7) $\displaystyle=$
$\displaystyle-\frac{d}{dt}\left(e^{-\int_{0}^{t}\rho_{D}(t^{\prime})dt^{\prime}}\right),$
(8)

and:

$\displaystyle\ln\left(1-P_{D}(t)\right)=$
$\displaystyle\int_{0}^{t}\rho_{D}(t^{\prime})dt^{\prime},$ (9)
$\displaystyle\Rightarrow\rho_{D}(t)=$
$\displaystyle\frac{p_{D}(t)}{1-P_{D}(t)},$ (10)

where in the left hand side of Eq. (9) we made use of the initial condition
$P_{D}(0)=0$.

As mentioned before, we assume that our creatures have a probability of dying
spontaneously due to aging. This occurs in such a way as to provide a lifetime
of $\lambda=1000$ time steps, and a slowly increasing $P_{D}(t)$. We chose a
logistic sigmoid function for this purpose:

$\displaystyle P^{\text{age}}_{D}(t)=\frac{1}{1+e^{-2(t-\lambda)/\lambda}},$
(11) $\displaystyle
p^{\text{age}}_{D}(t)=\frac{2}{\lambda}\frac{e^{-2(t-\lambda)/\lambda}}{\left(1+e^{-2(t-\lambda)/\lambda}\right)^{2}},$
(12)
$\displaystyle\rho^{\text{age}}_{D}(t)=\frac{2}{\lambda}\frac{1}{1+e^{-2(t-\lambda)/\lambda}}.$
(13)

Since individuals are also susceptible to dying from lack of energy, the total
rate of deadly events is
$\rho_{D}(t)=\rho_{D}^{\text{age}}(t)+\rho_{D}^{E}(t)$. We can relate it to
the age distribution in the population, which we will denote by $z(t)$, where
$z(t)dt$ is the amount of creatures of age $t$. If $N$ is the total
population, $\int z(t)dt=N$. The age distribution evolves according to:

$\displaystyle z(t+dt)$ $\displaystyle=z(t)-z(t)\rho_{D}(t)dt,$ (14)
$\displaystyle d_{t}z(t)$ $\displaystyle=-z(t)\rho_{D}(t),$ (15)
$\displaystyle\Rightarrow\rho_{D}(t)$
$\displaystyle=-d_{t}\ln\left(z(t)\right).$ (16)

Equation (16) expresses the total rate of deadly events in terms of age
distribution. Age distributions are shown in Fig. 5 in the steady-state of
evolution for three values of the resource recovery parameter $r$. Since they
are almost perfect exponentials, equation (16) implies that the death rate is
constant. Its inverse is the average overall lifetime of an individual in the
population. For all three cases, the obtained lifetime is $\tau=181\pm 9$. We
can therefore conclude that death is governed by lack of energy, not aging.
Lifetime is enough for growth and reproduction, but well under the
$\lambda=1000$ available if only death by old age is considered. Again, the
system has produced an altruistic behavior of sorts arising from selfish
rules.

Figure 5: Age distributions normalized to the total population in the
stationary regime of the system’s evolution. Since they are almost perfect
exponentials, the mortality rate given by using Eq. (16) is $\rho_{D}=0.055\pm
0.003$. Its inverse is the expected age, $\tau=181\pm 9$.

An interesting way to view individual lifetime is by looking at the
autocorrelation of the population. It can be obtained by applying the Wiener-
Khinchin theorem [20] to the power spectrum shown in Fig. 3. The result is
shown in Fig. 6. It is interesting to note how the correlation drops quickly
just above $t=100$, which we attribute to the fact that almost all creatures
in the simulation die in this range, and therefore the system loses coherence
throughout the creatures.

Figure 6: Population autocorrelation obtained from the population power
spectrum. Given the different population sizes for different values of $r$,
the autocorrelation is measured on deviations from the population average.

##  4 Mean-field dynamics

In this section we present a mean-field description of the resource dynamics,
validated with the results obtained in simulations.

The average consumption rate of a creature $c$ with genotype
$\mathbf{G}=\\{M,P,s\\}$ and phenotype $\mathbf{F}=\\{m,p\\}$ is $\langle
C\rangle=C(\mathbf{F}_{c})=\mu m^{\nicefrac{{3}}{{4}}}+\kappa
mp+sm\left(1-\nicefrac{{m}}{{M}}\right)$. For a population of creatures
$\mathcal{P}:c\in\mathcal{P}$ of which a subset $\mathcal{P}(\mathbf{x})$ are
at a site $\mathbf{x}$ at a certain time, their combined average consumption
during a short interval $\Delta t$, $a(\mathbf{x})$, is:

$a(\mathbf{x})=\min\left(\sum_{\mathcal{P}(\mathbf{x})}C(\mathbf{F}_{c})\Delta
t,f(\mathbf{x})-0.01K\right).$ (17)

Note that $\lim_{\Delta t\rightarrow
0}a(\mathbf{x})=\sum_{\mathcal{P}(\mathbf{x})}C(\mathbf{F}_{c})dt$ unless the
resource is strictly $0.01K$. Thus, in the continuous time limit, the average
resource dynamics can be obtained as follows:

$\displaystyle\frac{d\langle f\rangle_{\mathbf{x}}}{dt}$
$\displaystyle=\frac{1}{L^{2}}\sum_{\mathbf{x}}\left[rf(\mathbf{x})\left(1-\frac{f(\mathbf{x})}{K}\right)-\frac{a(\mathbf{x})}{dt}\right],$
$\displaystyle=\frac{1}{L^{2}}\sum_{\mathbf{x}}rf(\mathbf{x})\left(1-\frac{f(\mathbf{x})}{K}\right)-\frac{N}{L^{2}}\frac{1}{N}\sum_{\mathbf{x}}\sum_{\mathcal{P}(\mathbf{x})}C({\bf
F_{c}}),$ $\displaystyle=r\left[\langle
f\rangle_{\mathbf{x}}\left(1-\frac{\langle
f\rangle_{\mathbf{x}}}{K}\right)-\frac{\text{var}(f)}{K}\right]-\frac{N}{L^{2}}\langle
C\rangle_{\mathcal{P}}.$ (18)

In Eq. (18) we have assumed that $r$ and $K$ are homogeneous throughout the
landscape for simplicity, but of course this does not need to be the case, and
the averages should be taken accordingly. If we normalize the resource by its
local carrying capacity $K$, defining $n=f/K$, the dynamics becomes:

$\frac{d\langle n\rangle_{\mathbf{x}}}{dt}=r\left[\langle
n\rangle_{\mathbf{x}}\left(1-\langle
n\rangle_{\mathbf{x}}\right)-\text{var}(n)\right]-\frac{N}{L^{2}}\frac{\langle
C\rangle_{\mathcal{P}}}{K}.$ (19)

This equation resembles the prey’s dynamics in a predator-prey system, where
the normalized resource takes the role of the prey, and $N/L^{2}$ is the
normalized predator population. The interaction is a consumption of prey at a
rate given by the last term in Eq. (19), represented by $C$, since this comes
from the consumption rate $a$, which is explicitly related to $f$, the
“actual” prey.

Figure 7 shows both sides of Eq. (19) as measured from numerical simulations
of the system. Two simulations are shown, identical in all except the initial
population. The first one, denoted by $n_{1}$, had an initial population of
only $1$ individual. We observe that it starts and stays in equilibrium
($d_{t}\langle n\rangle_{{\bf x}}=0$). The second one, denoted by $n_{1000}$
has an initial population of $1000$ and achieves equilibrium after a transient
of $t\approx 1000$ time steps. After that, all curves follow similar paths. In
both cases the measured left and right-hand sides of Eq. (19) coincide almost
perfectly, supporting the validity of the mean-field dynamics. The initial
difference between the $n_{1000}$ curves are a product of different sampling
rate when measuring $d_{t}\langle n\rangle_{{\bf x}}$ and the corresponding
right-hand side.

Figure 7: Resource dynamics for different initial conditions. Both sides of
Eq. (19) are shown. Top: initial condition is a single individual. Bottom:
initial condition has 1000 individuals.

To complete the description of the dynamics and calculate the system’s
equilibria we need an equation for $d_{t}N$. The dynamics of $N$ is given by
the balance of births and deaths in the population. Since both of these are
controlled by the energy of the creatures, which fluctuates because of the
resource, births and deaths are stochastic processes. For this reason we must
first advance the understanding of the resource dynamics and its fluctuations
around its mean value.

###  4.1 Resource distribution

Figure 8: True and perceived resource distributions in the stationary state
for several values of recovery rate. Right: transformed distributions, to
remove divergences.

Figure 8 (left, thick lines) shows the resource distribution in equilibrium
for different values of $r$. Since the distributions have sharp spikes at both
extremes of the resource range, $0$ and $1$, we prefer to transform them by
taking the logarithm and multiplying by $n(1-n)$ to eliminate any possible
divergences. The right panel (thick lines) shows these transformed resource
distributions.

Since creatures need to be in a cell to consume its resource, the distribution
they perceive is the conditional distribution $\rho(n|\text{o})$ of resource,
given that the cell is occupied. Figure 8 (left, thin lines) shows the
corresponding perceived distributions as measured from the simulations (for
the same values of $r$). Although their concavities are different, as well as
their divergences, the perceived distributions display a shape similar to the
true ones when transformed accordingly, as shown in Fig. 8 (right, thin
lines).

The true and perceived distributions are related by consumption and recovery,
and in equilibrium we can derive their relation explicitly. First we decompose
the temporal evolution of $\rho$ according to its dependence on the resource
$n$:

$\frac{d\rho(n)}{dt}=\frac{d\rho(n)}{dn}\frac{dn}{dt}.$ (20)

Conditional probabilities allow us to write this in terms of occupied (o) and
not occupied ($\neg$o) sites, and equal to zero in equilibrium:

$\frac{d\rho(n)}{dt}=\frac{d\rho(n|\text{o})}{dn}\rho(\text{o})\left.\frac{dn}{dt}\right|_{\text{o}}+\frac{d\rho(n|\neg\text{o})}{dn}\rho(\neg\text{o})\left.\frac{dn}{dt}\right|_{\neg\text{o}}=0.$
(21)

Now, the dynamics of unoccupied sites is just the logistic recovery,
$d_{t}n|_{\neg o}=rn(1-n)$, while that of occupied sites is a balance of
recovery and consumption, $d_{t}n|_{o}=d_{t}n|_{\neg o}-\langle
C\rangle_{\mathcal{P}}/K$. Using these in Eq. (21), together with the fact
that $\rho(o)\rho(n|o)+\rho(\neg o)\rho(n|\neg o)=\rho(n)$, elementary algebra
allows to derive:

$rn(1-n)\frac{d\rho(n)}{dn}=\frac{\langle
C\rangle_{\mathcal{P}}}{K}\frac{N}{L^{2}}\frac{d\rho(n|o)}{dn}.$ (22)

Both sides of Eq. (22) are shown in Fig. 9, calculated from measurements
performed on simulations for the three values of $r$ shown in previous
figures. We can see that both bundles of curves coincide, within fluctuations,
for a very wide range of values of $n$, separating when $n$ approaches unity.
It is worth noting that Eq. (22) fails as $n\to 1$, which indicates that this
region never reaches equilibrium fully.

Figure 9: Dynamics of the resource, showing both sides of Eq. (22). For values
of $n\approx 1$, the curves separate, indicating that the system is not in
equilibrium in this region.

##  5 Discussion and conclusions

We have proposed and analyzed an agent-based model in which the energy
available to living creatures determines the processes that characterize life:
birth, reproduction and death. The dynamics of energy itself is defined by
genes and indirect competition with other creatures through a common resource.
Genes determine body size (i.e., mass and energy reservoir size), movement
speed across the domain of the resource and growth speed from birth to
adulthood. Small, random mutations upon asexual reproduction allow different
genetic combinations to compete with one another and therefore natural
selection emerges with only a shared resource and random mutations.

It is interesting to relate an intuitive notion of the intensity of natural
selection with the system’s capacity to sustain a population. In this sense,
natural selection is the force that eliminates individuals while the recovery
speed of the resource keeps them alive. As opposed to some artificial
selection models in which a quality of individuals is explicitly given and so
the best can always be picked, natural selection doesn’t guarantee a surviving
population. This is seen for simulations starting with a large genetic
diversity and small values of resource recovery speed. For larger recovery
speeds, an initial mass extinction is observed, followed by an explosion of
the successful genes. Even when the resource recovers instantly (i.e., the
resource is limitless), some genes are still unfit to survive and the initial
extinction is still observed. We conclude that natural selection decreases
with the capacity of a system but is always present to some degree.

After the initial extinction and population explosion, we found that two
distinct genetic groups emerge in the stable state of the system. They are
separated by their moving speed but indistinguishable in their mass. The
growth speed, on the other hand, turned out to be a gene with which fitness
increased monotonously, so it never reached a stationary value but didn’t
affect the system. Since a faster movement implies more energy consumption,
fast populations were found to be smaller. However, only one of these
“species” survived in each realization. Species were observed to compete
briefly in the initial transient period, as shown in Fig. 10, but one of the
groups always eliminated the others in accordance with the competitive
exclusion principle [21] and punctuated equilibrium.

Figure 10: Brief coexistence of several distinct species in the initial,
transient regime of a typical simulation. The recovery speed is $r=0.0175$ and
the state occurs about halfway through the transient period.

Simulations starting with a single individual stayed in equilibrium throughout
the evolution of the system, even during population growth. They also
exhibited stationary states at resource recovery speeds slower than ones at
which larger populations completely died out. Additionally, these populations
always converged to low energy consumption populations. This suggests that the
high energy consumption species found earlier are a result of a non-
equilibrium situation, and thus the system is very sensitive to initial
conditions. Another noteworthy conclusion is that, according to this model,
life beginning from a single organism seems to be more likely than life
beginning from a larger group, even at the cost of genetic diversity.

The final result we would like to remark from simulations is that, even though
the rules followed by each individual were set to be selfish, the system
managed to evolve into a somewhat altruistic setting. This was observed in two
instances. The first regards the amount of resource consumed, where creatures
were supposed to eat enough to fill their energy reservoirs but ended up
keeping them at the minimum necessary to reproduce. The second relates to
lifetime, that could end by old age after around ten reproductions, but ended
up limited to the time necessary for just one reproduction. This makes perfect
sense because the population must remain constant, but it also shows that the
probability of producing offspring, usually associated with fitness, is not
maximized under natural selection with a shared resource. The population size,
used as a fitness measure in works such as [9, 10] does not seem to be
maximized either, since for the same conditions we have found realizations
that attain a stationary state with a large population and low consumption,
and others with small populations and high consumption. It is worth
emphasizing that this single-offspring situation was not programmed in the
model, but emerged from constraints and interactions.

Although the model is somewhat artificial (agents are, quite literally,
spherical cows), our study demonstrates that all of the phenomena mentioned
above are actually parts of a whole. They can be obtained by considering that
the genetic code of an individual determines how energy is obtained and
employed to harvest more. An open question is whether more detailed models of
genetic dependence produce more realistic results in terms of surviving genes.
Another is if predator-prey systems are equivalent to resource-population
systems where the resource is controlled by genes in the same way as the
population.

We have taken an initial step towards answering the first question by adding a
rudimentary intelligent behavior to the population. With it, movement speed
depends linearly on the amount of energy in a creature’s reservoir instead of
being constant. This allows creatures to move more when they are hungry and
less when they are full and vice-versa. The first case is more energy
efficient but dangerous when there is little food nearby, while the second is
less efficient but could be safer in terms of keeping energy levels high. Both
cases have their advantages and drawbacks and therefore we cannot predict
which will appear. However, it is known that the flatworm Dugesia tigrina
exhibits this rudimentary intelligence and behaves as in the first case [22].
In our simulations, we found that the creatures in the stationary state were
those with genes that made them move more when they were hungry and less when
they were not. Since this agrees with the real-life scenario we conclude that,
as genetic models get better, the simulation results will also get more
realistic.

Finally, we would like to mention the consequences of the mean-field dynamics
analysis. Much can be read from Eq. (19), describing the dynamics of the mean
normalized resource, which we copy here:

$\frac{d\langle n\rangle_{\mathbf{x}}}{dt}=r\left[\langle
n\rangle_{\mathbf{x}}\left(1-\langle
n\rangle_{\mathbf{x}}\right)-\text{var}(n)\right]-\frac{N}{L^{2}}\frac{\langle
C\rangle_{\mathcal{P}}}{K}.$ (23)

First, Eq. (23) resembles the dynamics of a prey, $\langle n\rangle$, faced
with a population of $N$ predators consuming them at a rate $\langle
C\rangle/KL^{2}$, as mentioned above. This interpretation suggests that our
model is similar to a predator-prey system. It also makes altruistic behavior
(keeping energy at the minimum necessary) beneficial not only for the rest of
the population (other predators), but also for the resource (the “prey”). Of
course, this is a consequence of the population needing the resource (or
predators needing prey) to survive. However, since we observed populations
with higher energy consumption, we can conclude that this form of altruism
isn’t necessary, merely better.

Second, note that the growth term of the resource in Eq. (23) is reduced by
the variance of the resource. This means that smaller fluctuations in the
resource distribution at the cost of a smaller expected value, sometimes will
be more beneficial for the population. This effect is definitely non-
negligible since an estimation of the population that does not include this
term, $N\approx rf(1-f)L^{2}/C\approx 500$ ($r=0.01$, $P=0.33$, $f=0.85$),
results in an error of around $70\%$ compared to the observed population of
$N\approx 300$.

Finally, since all real-world food sources have a finite size, their growth is
limited. Therefore a logistic equation, as a second-order expansion, is the
simplest realistic assumption we can make. The variance term in Eq. (23) stems
from the nonlinear term in this expansion and therefore all real-world
resources will suffer from it. Moreover, resource distribution is shaped by
consumption, so the efficiency of a species can be characterized through that
variance. In this sense, more “intelligent” (better distributed) consumption
will lead to faster resource recovery and therefore a larger population. This
may explain why the first great revolution human beings experienced, the
invention of agriculture, was so important [23, 24]. Since we could suddenly
control the dynamics of our source of energy, we could minimize variance much
more effectively, ensuring that everyone has enough food and allowing the
population to grow.

We have set out trying to describe a model that produces natural selection
from first principles, namely energy conservation and its flow through living
beings. In the process, we have discovered many other known results and
characteristics associated with the theory of evolution. We consider this a
successful feat, but there is still much work to be done. It is not clear how
exactly these phenomena emerge from the underlying process, nor how they are
influenced when specifics are changed. We hope to address these questions and
any new ones that may appear in future work.

## Acknowledgments

We recognize partial support through the following grants: CONICET PIP
112-2017-0100008 CO, UNCUYO SIIP 06/C546, ANPCyT PICT-2018-01181. We also
thank S. Bouzat for fruitful discussions.

## References

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  * [2] Eric Smith and Harold J. Morowitz.  The origin and nature of life on Earth.  Cambridge UP, 2016.
  * [3] Stuart A. Kauffman.  The Origins of Order: Self-Organization and Selection in Evolution.  Oxford University Press, 1993.
  * [4] Per Bak and Kim Sneppen.  Punctuated equilibrium and criticality in a simple model of evolution.  Physical Review Letters, 71:4083, 1993.
  * [5] Guillermo Abramson and Damián Zanette.  Statistics of extinction and survival in lotka-volterra systems.  Physical Review E, 57:4572, 1998.
  * [6] Samira El Yacoubi and Sebastien Gourbière.  A cellular automata model for adaptive sympatric speciation.  In Samira El Yacoubi, Bastien Chopard, and Stefania Bandini, editors, Cellular Automata, volume 4173 of Lecture Notes in Computer Science, pages 287–296. ACRI 2006, Springer, 2006.
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  * [14] Max Kleiber et al.  Body size and metabolism.  Hilgardia, 6(11):315–353, 1932.
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  * [19] Richard Dawkins.  The selfish gene.  Oxford UP, 1976.
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  * [21] James D. Murray.  Mathematical biology.  Springer, 2002.
  * [22] I. Sheiman, E. Zubina, and Natalia Kreshchenko.  Regulation of the feeding behavior of the planarian _Dugesia (Girardia) tigrina_.  Journal of Evolutionary Biochemistry and Physiology, 38:414–418, 07 2002.
  * [23] Vered Eshed, Avi Gopher, Timothy B Gage, and Israel Hershkovitz.  Has the transition to agriculture reshaped the demographic structure of prehistoric populations? New evidence from the levant.  American Journal of Physical Anthropology: The Official Publication of the American Association of Physical Anthropologists, 124(4):315–329, 2004.
  * [24] Graeme Barker.  The agricultural revolution in prehistory: why did foragers become farmers? Oxford University Press on Demand, 2009.


# On Einstein’s last bid to keep a stationary cosmology

Salvador Galindo-Uribarri∗ and Jorge L. Cervantes-Cota∗∗ Departamento de
Física, Instituto Nacional de Investigaciones Nucleares,
Carretera México Toluca Km. 36.5, Ocoyoacac, C.P. 52750, Edo. Mex., México.
∗He recently passed away.
∗∗E-mail<EMAIL_ADDRESS>

###### Abstract

It is commonly known that the steady-state model of the universe was proposed
and championed in a series of influential papers around mid-twenty century by
Fred Hoyle, Hermann Bondi, and Thomas Gold. In contrast it is little known
that, many years before, Albert Einstein briefly explored the same idea; that
is of a “dynamic steady state” universe. In 1931 during his first visit to
Caltech, Einstein tried to develop a model where the universe expanded and
where matter was supposed to be continuously created. This latter process was
proposed by him to keep the matter density of the universe constant. However,
Einstein shortly abandoned the idea. The whole event has already been
described and analyzed by C. O’Raifeartaigh and colleagues in 2014. It is the
purpose of this brief note to point out what might have prompted Einstein to
consider a continuous creation of matter and the prevailing circumstances at
that time that drove Einstein’s intent.

###### keywords:

History of Physics; Cosmology; Einstein.

##  1 Introduction

The first relativistic model of the cosmos is due to Einstein. His model
entailed the idea of a universe both isotropic and homogeneous on the largest
scales (idea known as “Cosmological Principle”). At that time, it was the
accepted view that the universe was stationary. This was not unreasonable
since relative velocities of stars are small. To achieve a stationary
universe, Einstein added in 1917 to his original field equations, the so-
called cosmological constant “$\lambda$” to counterbalance the effects of
gravity and attain a static universe[1]:

$G_{\mu\nu}-\lambda\,g_{\mu\nu}=-\kappa\left(T_{\mu\nu}-\frac{1}{2}g_{\mu\nu}T\right)\,.$
(1)

Einstein showed that the value of the constant he introduced was proportional
to the mean mass density $\rho$ of the universe and inversely proportional to
the square of $R$, its radius of curvature,

$\lambda=\frac{\kappa\rho}{2}=\frac{1}{R^{2}}\,.$ (2)

The introduction of $\lambda$ in the original field equations was accepted by
the early few practitioners of General Relativity as convenient to keep the
current standard view at the time, of a static universe.

However, Einstein was uneasy with his forced introduction of the $\lambda$
constant. Several of his papers show his uneasiness. For instance, in a 1919
publication Einstein stated:

“…the general theory of relativity requires that the universe be spatially
finite. [This] requires the introduction of a new universal constant
$\lambda$, standing in a fixed relation to the total mass of the universe (or
respectively, to the equilibrium density of matter). This is gravely
detrimental to the formal beauty of the theory” [2] .

It was not until 1931 that Einstein at long last dropped off $\lambda$ from
his field equations. During those days, Einstein was visiting professor in
Pasadena California attending an invitation made by Robert Andrews Millikan to
spend a short season at Caltech (from late December 1930 to early March 1931).
Soon after his arrival to California he discussed Hubble’s redshift
measurements and its implications with Mount Wilson Observatory astronomers.

Today there is still a justified widespread view that Einstein discarded the
cosmological constant immediately after he was satisfied of the validity of
Hubble’s evidence for a non-static universe. This rendered $\lambda$ in his
field equations, redundant. In April of that same year, Einstein submitted for
publication, a paper to the Sitzungsberichte der Preussischen Akademie der
Wissenschaften where he rejected his cosmological term as superfluous and no
longer justified. In his own words, “theoretically unsatisfactory” [3].

But the view that Einstein dropped off $\lambda$ just before his meetings with
the Mount Wilson astronomers (in early Jan 1931) is not strictly exact. In the
interval between his arrival to California and his 1931 Sitzungsberichte paper
(April 1931), he made a last effort to model a “dynamic steady state”
universe, keeping $\lambda$ in his field equations. This was serendipitously
discovered in 2013 by Cormac O’Raifeartaigh in an unpublished Einstein’s
manuscript kept in the Albert Einstein Archives (AEA) maintained by the Hebrew
University of Jerusalem[4]. The first translation into English of the
manuscript, contents and its analysis, has been already covered in 2014 by his
discoverer Cormac O’Raifeartaigh and colleagues [5]. In addition, the
manuscript contents were also commented in a note added in proof by Harry
Nussbaumer and later also reviewed by him [6, 7].

##  2 A significant Finding

The discovery by O’Raifeartaigh of the unpublished manuscript showing a model
still using the lambda constant, came as a great surprise to everyone. From
the moment Einstein arrived at Caltech it seemed that he had already accepted
a non-static universe. Various reports in the local press affirm this. And
consequently, people supposed that Einstein had already scrapped the constant
from his field equations as unnecessary.

As a celebrity, during his visit to California, Einstein’s activities and
sayings were reported daily by the press. In Jan 3 the New York Times (NYT)
reported that during an interview given the previous day Einstein stated: “New
observations by Hubble and Humason (astronomers at Mount Wilson) concerning
the red shift of light in distant nebulae make the presumptions near that the
general structure of the universe is not static” [8]. The next following month
the NYT (Feb 5) another front-page story, informed that Einstein delivered a
lecture the previous day (Feb 4) where he no longer held to the model of
“Stable universe”[9].

On January 15, Einstein attended a welcome dinner in his honor at “The
Athenaeum”, a club house for the California Institutes Associates (a group of
promoters of southern California scientific and scholarly research). The
dinner was attended by around 200 guests among them a selected group of
astronomers that had collaborated with their own research in testing
relativity[10]. The final dinner speech was delivered by one of them, Walter S
Adams, director of Mount Wilson. Adams had verified Einstein’s prediction of
gravitational redshift[11]. During his speech Adams highlighted the problem of
the “nature and structure of the universe” and he announced that:

“Professor Einstein is now inclined to consider the most promising line of
attack on the problem to be based on theories of a non-static universe, the
general equations of which have been developed so ably by Dr. Richard Chase
Tolman, of the California institute of technology” [12].

Despite the multiple examples that can be cited, affirming Einstein’s alleged
intention to follow a non-static approach, the 2014 discovery by
O’Raifeartaigh of the manuscript, revealed that Einstein temporarily followed
a different scheme in his unpublished document.

##  3 The manuscript

The unpublished manuscript is entitled “Zum kosmologischen Problem”, that is
located in the Albert Einstein Archives (AEA) (draft, 1931. Doc [2-112]). It
is a signed, four-page handwritten manuscript by Einstein on American paper.
Assigned by AEA to January or February 1931.

The cosmological model depicted in the manuscript was not previously detected
given that the first words of its title are identical to those of Einstein’s
1931 Sitzungsberichte paper (April 1931). For this reason, it was assumed to
be a draft of the latter publication.

The manuscript has already been extensively analyzed by O’Raifeartaigh and
colleagues [5] and in the here cited papers by Nussbaumer, so we shall limit
ourselves to giving a succinct description of its contents based on those
publications.

##  4 The model in the manuscript

In the manuscript Einstein explores a solution to his field equations
retaining $\lambda$ that could be compatible with Hubble’s observations; to be
precise an expanding universe in which the density of matter does not change
over time. Einstein starts his analysis by choosing the metric of flat space
expanding exponentially (De Sitter metric):

$ds^{2}=-e^{\alpha t}(dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2})+c^{2}dt^{2}.$ (3)

So, the distance between two points increases over time as $e^{\alpha t/2}$,
and he remarks: “one can thus account for Hubbel’s [sic] Doppler effect by
giving the masses (thought of as uniformly distributed) constant co-ordinates
over time”.

In his calculations he finds that $\alpha^{2}$ (representing the expansion of
the universe) is related to its overall density $\rho$ as,

$\alpha^{2}=\frac{\kappa}{3}\rho.$ (4)

That is, $\rho$ determines the expansion. Since the redshift measurements at
the time suggested that the universe expansion is constant, so he concluded
that the density must be constant as well.

In the final part of his manuscript, Einstein proposes that the density of
matter remains constant by supposing a continuous formation of matter in empty
space. He then observes that: “For the density to remain constant, new
particles of matter must be continually formed within that volume from space.”

At the end of the manuscript, he associates the cosmological constant with an
energy of space: “by setting the $\lambda$-term, space itself is not empty of
energy”. Consequently, the continuous creation of matter becomes associated to
the $\lambda$ cosmological constant.

Einstein’s “dynamic steady state” universe simultaneously incorporated the
observed expansion of the universe together with a new paradigm, namely of an
expanding universe that maintained its isotropy and spatial homogeneity the
same as it always has and always will (later dubbed the perfect cosmological
principle) by means of the continuous creation of matter.

The “dynamic steady state” model was so compelling to set aside. However,
Einstein didn’t submit his manuscript for publication. O’Raifeartaigh and
colleagues detected the possible reason why Einstein gave up publishing it[5].
He must have noticed on revision, that his model contained a flaw. Once the
error is corrected the model leads to the trivial solution $\rho=0$, that is
an empty universe of matter. Therefore, he promptly abandoned this attempt.

We must ask now what it was that attracted Einstein to consider the
possibility of a universe where there is a continuous creation of matter. To
find a possible answer we shall consider the reception at that time, of an
expanding universe as opposed to prevailing static views.

##  5 Universe’s age and Earth’s age tension

Towards the beginning of the 1930s the main objection to the interpretation of
the Hubble constant as the indicator of the universe expansion was that it’s
measured value implied a younger universe than that of Earth’s accepted age.

Before 1930’s, scientific estimates of the age of the Earth dated from
mid-19th century, from Helmholtz-Kelvin gravitational contraction ages[13],
passing through those resting on geology (i.e., the amount of salt in the
oceans, sediments) and - after a fierce debate between geologist and
physicists[14] \- finally deferred to an age based on early-20th century
radioactive dating[15]. This produced by the 1930’s, an estimate of the
universe age of around 4 billion years[16].

In 1931, at the time Einstein was visiting Caltech, the matter of determining
the age of the earth was not entirely resolved to the extent that the National
Research Council decided to appoint a committee to investigate to settle the
question[17].

On the other hand, the cosmological age of the universe based on the universe
expansion, was estimated by taking the reciprocal of the value of the Hubble
constant (Hubble’s time). Hubble first evaluation of his eponymous constant
was of around 500 kilometers per second per megaparsec. This implied a
universe’s age of about two billion years, which was in a tense contradiction
with the estimated age of the Earth, as just mentioned, of about four billion
years. As consequence, such mismatch created room for doubt, Einstein himself
included. Some critics questioned that the observed nebulae redshifts, were in
fact a manifestation of the Doppler’s effect. Such was the panorama that
reigned in the early 30s.

This incongruity raised two possible explanations: on the one hand Hubble
constant value was wrong, or on the other hand, Doppler’s shift needed a novel
interpretation, as it is said today “perhaps new physics”.

As it is well known, the Hubble constant estimation involved measurement of
distance and velocity of a variable star (a Cepheid) belonging to a galaxy in
question. This involved monitoring the apparent brightness of the Cepheid
variable to obtain its period. Then, using Shapley’s Cepheid calibration, its
absolute brightness was established and consequently the distance to the
galaxy where the Cepheid was located. At that time there was little doubt on
the correctness of Shapley’s calibration. Also, measurement of recession
velocity of galaxies even in those days, was straightforward. Spectral
recordings had already achieved good accuracy. So, Hubble’s constant value was
in little doubt. Eventually, some astronomers pointed out that Cepheid’s
calibration could be slightly inaccurate as the apparent star’s luminosity
could be diminished by interstellar media and that had to be accounted for. It
must be remembered that It was not until mid-1940’s that Walter Baade
identified two Cepheid types and made a major correction to Shapley’s
calibration and thus to Hubble’s value[18]. Now we know that the problem was
on the way off value of Hubble’s constant at that time.

On the second possibility, that the incongruity between the ages of the earth
and the universe had its origin in an erroneous interpretation of the observed
redshift, Fritz Zwicky gave in 1929 an alternative explanation. Zwicky, a
resident scholar at Caltech, suggested the concept of “tired light” [19]. This
was a hypothetical redshift mechanism where photons lost energy over time
through interactions with other particles in their trajectories through a
static universe. So, the more distant objects would appear redder than more
nearby ones. It is pertinent to mention that the term ”tired light” was not
used by Zwicky but later coined to refer to this concept by Caltech
cosmologist Richard Tolman in the early 1930s in relation to the so called
Tolman Surface Brightness Test.

##  6 Earlier dynamic stationary state cosmologies.

Zwicky was not alone. There was also an academic minority that put forward
similar ideas to Zwicky’s explanation even before his 1929 suggestion. All
these “tired light” propositions had in common the assumption that photons
heading earthward, somehow interchanged with the intergalactic medium, part of
their energy, thus red shifting their frequencies.

“Tired light” was championed by the few enthusiasts of steady state
cosmologies. Its appeal as we shall see resides on the fact that it
circumvents the standard interpretation of the redshift as an indicator of
galaxy recession and thus safeguards their view of a static universe.

The Nobel laureate Walther Nernst and William Duncan Macmillan, a well-known
Chicago professor, were leading advocates of a steady-state universe. What
these had in common is that they presuppose the universe as eternal, self-
preserving structure in which matter and radiation are constantly been
transformed into one another to keep the universe in a stationary balanced
state thus avoiding a heat-dead of the universe. A cosmic destiny they both
abhorred.

MacMillan’s supposes in his own version of “tired light”: “That there is a
leakage of energy from the photon […] due perhaps to an inherent instability
in the photon, or, possibly, [due] to collisions with other photons”. He
concludes therefore: “it is evident that the frequency [of the photon]
declines with the energy, and the lines of the spectrum are shifted toward the
red” [20].

In addition, he considers that the “evaporated” energy from the photon
continues to exist as abundant radiant energy of “very low frequency”. A kind
of primitive version of the modern CMB radiation (but the latter being Black
Body radiation). Then he puts forward the possibility that perhaps the
“evaporated” energy “disappears into the fine structure of space and reappears
eventually in the structure of the atom”. In other words, he proposes a
mechanism of matter creation.

Another notable hypothesis was that of the Nobel laureate, Walther Nernst, who
suggested that some photons are partially absorbed by the luminiferous
$\ae$ther (which he accepts to exist)[21]. In his 1928 essay “Phyco-Chemical
Considerations in Astrophysics”, Nernst outlines the notion of matter creation
in his stationary universe:

“I may therefore hold fast to the hypothesis uttered by me that, just as the
principle of the stationary condition of the cosmos demands that the radiation
of the stars be absorbed by the luminiferous $\ae$ther, so also finally the
same thing happens with mass, and that, conversely, strongly active elements
are continually being formed from the $\ae$ther, though naturally not in
amounts demonstrable to us,…” [22].

It is worth noticing that both, MacMillan and Nernst, claim energy transfer
occurs by an unknown process yet to be discovered.

The “tired light” proposal did not vanish into oblivion as alternative
explanation to redshift been caused by recessional motion. As late as 1935,
Edwin Hubble himself and Richard Tolman (both at Caltech during Einstein’s
several yearly visits) investigated the possibility that “tired light” might
be an alternative interpretation of redshift.

They did that comparing the surface brightness of galaxies as a function of
their redshift (applying the so called “Tolman Surface Brightness Test”[23]).
According to Tolman Test, the relationship between surface brightness of a
galaxy and its redshift differs in the case of a static universe from that of
an expanding one. In a coauthored publication they explored this
possibility[24]. In their joint paper and to simplify their analysis, they
employed Einstein’s static model of the universe: … “combined with the
assumption that the photons emitted by a nebula [a galaxy] lose energy on
their journey to the observer by some unknown effect, which is linear with
distance, and which leads to a decrease in frequency, without appreciable
transverse deflection”. In short, they employed Einstein’s model plus “tired
light”. As result of their analysis, they conceded: “Until further evidence is
available, both the present writers wish to express an open mind with respect
to the ultimately most satisfactory explanation of the nebular red-shift”
…“They both incline to the opinion, however, that if the red-shift is not due
to recessional motion, its explanation will probably involve some quite new
physical principles”.

##  7 On Einstein’s arrival at Caltech

Up until now, we have seen that in the late 1920s and early 1930s, Hubble’s
law was commonly interpreted as a demonstration that the universe was truly
expanding. However, there was still the problem of reconciling the age of the
universe with that of the earth. This inconsistency, in addition to raising a
reasonable doubt about the interpretation of the redshift, gave rise to the
credibility of the theories of a stationary universe. For some, the appeal of
a universe without evolving through time resided in that such universe has no
beginning and no end, so it converts the “age of the universe” as a senseless
question issue, consequently eliminating the earth-universe ages paradox.

Attempting to develop a stationary cosmological model was indeed an attractive
motivation for Einstein, since it would avoid his well-known aversion to a
universe that has a beginning and reinforce his well-known paradigm of a
static universe. But adopting such an idea required reliable and robust
observational evidence of the continuous creation of matter.

As we will see below, fresh “evidence” (that turned out to be illusory) of the
continuous creation of matter emerged just as Einstein arrived at Caltech.
This came from Millikan (Einstein host at Caltech) so it must have seemed
reliable to Einstein. We must remember that Millikan received the Nobel prize
partly for experimentally verifying Einstein’s photoelectric equation.

##  8 Birth cries of atoms

On the 1st of February 1931, during Einstein’s stay at Caltech, a paper by
Millikan and his collaborator H. Cameron appeared in Physical Review. Its
title was “A more accurate and more extended cosmic-ray ionization -depth
curve, and the present evidence for atom -building”[25]. This paper contained
what it would seem to be the experimental evidence Einstein needed, that is,
observational evidence of continuous matter creation. In our opinion, it was
this publication what encouraged Einstein to use the continual creation of
matter hypothesis in his unpublished manuscript. So, Einstein made his last
bid to keep a stationary cosmology.

Millikan’s 1931 paper is the apogee of a series of publications on cosmic rays
where he explores their composition and origins. But, before we comment on
this publication it is convenient to recall very briefly Millikan’s research
on cosmic rays. During the early 20’s he and his collaborators made a series
of observations on board of balloons, high altitude peaks, and different
geographical locations, at various latitudes and in the depths of lakes
proving in his 1926 publication that cosmic rays were of extraterrestrial
origin[26]. Millikan believed that his measurements proved that the primary
cosmic rays were energetic photons. He also stated that cosmic rays, “…must
arise from nuclear changes of some sort…” but far more energetic than any
radioactive change thus far on record”.

In the same 1926 paper he suggested that cosmic rays probably came from among
the following three nuclear processes: a) The capture of an electron by the
nucleus of a light atom, b) the formation of helium out of hydrogen, or c)
some new type of “nuclear change”, such as the condensation of radiation into
atoms.

Regarding third suggestion he made above, we must emphasize that Millikan here
raises the creation of matter from the “condensation of radiation into atoms.”
this idea was not his, as he himself stated in a note written by him for the
journal Science in 1930[27]. In the note he recalls having in 1915 discussions
on the “running down of the universe” (i.e., when the universe has reached a
state of maximum entropy) at the University of Chicago with his then colleague
William Duncan Macmillan (of whom we have already commented) where the issue
of atom building was discussed. Millikan recalls,

“In our conversations at Chicago W. D. Macmillan constantly held out for the
view that a still further step forward should be taken and that the idea of
“running down of the universe” should be given up by the assumption that atom
building went on in space by the condensation of radiation into atoms. He
discussed his idea in detail with me in the year 1915, and in July, 1918 he
published it in full”[28].

In 1928 Millikan and Cameron found that incoming cosmic rays could be grouped
into three independent energy bands centered at 26, 110, and 120 (MeV)[29].
They argued that these bands are produced by the release of photons when
eventually a “sudden union” (i.e., fusion) between atoms occur. To support
their assertion, they observed that the center of the bands agreed
respectively with 26 MeV (which is just about the mass defect of helium), 110
MeV (which is close to the mass defect of oxygen and nitrogen) and 220 MeV (to
that of silicon.) So, their inference was that the three photon bands reaching
the earth must be generated by the “sudden union” of: 4 hydrogen atoms fused
to form helium, 14 to form nitrogen, 16 to form oxygen and 28 to form
silicon”. Secondary electrons, they claimed, were produced in the atmosphere
by Compton scattering of gamma rays. Millikan believed that space was filled
with a tenuous gas of electrons and protons (the latter he called “positive
electrons”). So, to get around the “running down of the universe” he assumed
that “These building stones [protons and electrons] are continuously being
replenish throughout the heavens by the condensation with the aid of some as
yet wholly unknown mechanism of radiant heat into positive and negative
electrons”.

Millikan supposed discovery of the continuous formation of matter went beyond
the scientific sphere provoking the attention of among the scientific and the
lay publics. In a statement made by Millikan to the press, cosmic rays were
“birth cries of atoms, a Millikan phrase that achieved a good deal of
currency”[30].

Regarding Millikan’s 1931 publication the one which appeared on print just at
the time Einstein was hosted by Millikan, this was basically a continuation of
his 1928 paper including revised energy calculations that further bolstered
his tenacious ideas. A short time later Arthur Holly Compton showed that not
all cosmic rays were photons, but at least a large part of them consists of
charged particles. This led to a debate between the two Nobel laureates. The
debate went on for some time with Compton at the winning side but that’s
another story.

##  9 Final comments

Einstein before long abandoned the idea of keeping the $\lambda$ constant in
his field equations. In his next publication, the 1931 Sitzungsberichte paper,
he does not make use of the constant anymore[3]. At the end of the paper
Einstein adds some remarks about the age of the universe problem, which was
quite severe without the use of the $\lambda$ constant. Today we know that the
problem was on the way off value of Hubble’s constant at that time. Einstein
signals two possible errors that may be occurring in the initial approach to
the problem. First, he insinuates that the matter distribution might be
inhomogeneous and that a homogeneity approximation may be illusional. Then he
adds that in astronomy one should be cautious with large extrapolations in
time. His first comment is surprising as it seems to indicate some
reservations on the cosmological principle.

The next following winter (1931-1932), Einstein was back in Caltech for a
second stay where he met De Sitter. Together they formulated a model today
known as the Einstein-de Sitter universe, assuming a flat space with no
cosmological constant and with an expansion velocity asymptotically
approaching zero in the infinite future [31]. This became the standard model
up to the mid-1990s. This was the Einstein’s last intent to produce a
cosmological model.

## Acknowledgments

JLCC acknowledges support by CONACyT project 283151.

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