# Hardware Random number Generator for cryptography

Ram Soorat111corresponding author<EMAIL_ADDRESS>Madhuri K. and
Ashok Vudayagiri  School of Physics, University of Hyderabad, Hyderabad
500046, India

###### Abstract

One of the key requirement of many schemes is that of random numbers. Sequence
of random numbers are used at several stages of a standard cryptographic
protocol. A simple example is of a Vernam cipher, where a string of random
numbers is added to massage string to generate the encrypted code. It is
represented as $C=M\oplus K$ where $M$ is the message, $K$ is the key and $C$
is the ciphertext. It has been mathematically shown that this simple scheme is
unbreakable is key K as long as M and is used only once. For a good
cryptosystem, the security of the cryptosystem is not be based on keeping the
algorithm secret but solely on keeping the key secret. The quality and
unpredictability of secret data is critical to securing communication by
modern cryptographic techniques. Generation of such data for cryptographic
purposes typically requires an unpredictable physical source of random data.
In this manuscript, we present studies of three different methods for
producing random number. We have tested them by studying its frequency,
correlation as well as using the test suit from NIST.

##  I Introduction

A random process is a repeating process in which output is difficult to find a
describable deterministic pattern. The term randomness is quite often used in
statistics to signify well defined statistical properties, such as
correlation. Saying that a variable is random means that the variable follows
a given probability distribution. In such a definition, random is different
from arbitrary. A good RNG should work efficiently, which means it should be
able to produce a large amount of random numbers in a short period of time.
Random numbers are widely used in many applications, such as cryptography bb84
; crypto , spread-spectrum communications pickholtz , Monte Carlo numerical
simulations metro , and ranging naza  statistical analysis, numerical
simulations, information security, stochastic simulation, stream ciphers,
ranging signal in radar system, controlling signal in remote control,
encryption codes or keys in digital communication, address codes and spread
spectrum codes in code division multiple access (CDMA) and many others. So
simulations of random numbers are crucial. amounts of random numbers are
necessary and thus fast RNGs are required. They are also used in the
statistics to solve problems in many fields such as nuclear medicine, finance
and computer graphics.

There are in general two types of generators for producing random sequences:
true random number generators (TRNGs) and pseudo random number generators
(PRNGs). PRNGs need some input called seeds, along with some deterministic
algorithms to generate multiple pseudo random numbers. They are usually faster
than TRNGs and are preferable when a lot of random-like numbers are required.
Since traditional random numbers generated by algorithms are essentially
pseudo-random, they have potential danger in security-related fields like
quantum key distribution. TRNGs make use of non-deterministic sources along
with some post-processing functions for generating randomness. Such sources
include physical phenomena such as thermal noise, atmospheric noise,
radioactive decay and even coin tossing. such as electrical noises petrie ,
frequency jitters in electrical oscillators bucci  and chaotic circuits stoj ;
stoj1 , which can produce unpredictable random numbers of high quality yet
much lower rates than PRNGs because of the narrow bandwidth of these physical
entropy sources. In addition a number of documents exist which provide general
advice on using and choosing random number sources callas ; tim ; crypto1 ;
klaus . Further discussions on the nature of randomness, pseudo random number
generators (PRNG’s), and cryptographic randomness are available from a number
of sources donald ; paul ; crypto2 .

A true Random numbers is the base of many cryptographical applications like
QKD, especially to generate keys that cannot be penetrated by hackers or other
attackers it is important that the random numbers used is unpredictable. The
BB84 protocol makes use of polarization states of single photons to map the
bits 0, 1 of the encryption key, in two mutually unbiased basis. This results
in pulses containing single photons, each randomly this means that the RNG has
to perform better than the Pseudo random number generators (PRNG) available on
the computer, but also should be compact and easy to integrate into the
prototype QKD device. This requires a controller that generates four random
states and its deterministic critically endangers the security of the entire
protocol. For most applications it is desirable to have fast random number
generators (RNGs) that produce numbers that are as random as possible.

##  II Test for Randomness

There are different types of statistical tests that can be applied to a
sequence to attempt to compare and evaluate the sequence to a truly random
sequence. Random sequence can be characterized and described in terms of
probability. In addition, the results of statistical testing must be
interpreted with some care and caution to avoid incorrect conclusions about a
specific generator. We use a set tests designed and implemented by national
institute of standard and technology (NIST) test suite is a statistical
package consisting of 15 tests andrew . This package will address the problem
of evaluating (P)RNGs for randomness. It will be useful in

  * •

Identifying (P)RNG’s which produce weak (or patterned) binary sequences,

  * •

Designing new (P)RNG’s,

  * •

Verifying that the implementations of (P)RNG’s are correct,

  * •

Studying (P)RNG’s described in standards, and

  * •

Investigating the degree of randomness by currently used (P)RNG’s.

###  II.1 P-value

The probability (under the null hypothesis of randomness) that the chosen test
statistic will assume values that are equal to or worse than the observed test
statistic value when considering the null hypothesis. The P-value is
frequently called the ”tail probability”.

If a P-value for a given run equals to 1, then the sequence is decided to be
perfectly random. A P-value of zero indicates a completely non-random
sequence. In addition a significance level ($\alpha$) can be chosen such that
if $P-value\geq\alpha$, then the sequence can be considered to be almost
random. If $P-value<\alpha$, the sequence appears to be non-random. The
parameter $\alpha$ denotes the probability of the Type I error and typical
values for $\alpha$ is chosen in the range [0.001, 0.01].

An $\alpha$ of 0.001 indicates that one would expect one sequence in 1000
sequences to be rejected by the test if the sequence was random. For a
$P-value\geq 0.001$, a sequence would be considered to be random with a
confidence of 99.9%. For a $P-value<0.001$, a sequence would be considered to
be non-random with a confidence of 99.9%.

##  III  Generation of Random Numbers

A Random numbers are important requirement for both classical encryption as
well as QKD. Several attempts were made for random number generator, which
would be used for the entire QKD scheme. We present here our studies of three
different RNGs,

  * •

A hardware generator built around Kuuselas chaos circuit

  * •

Dark Counts from an Avalanche Photo Diode

  * •

Pseudo Random code from internal modules of LabVIEW

In all these methods we show the efficiency of random number generator by
analyzing their statistics.

##  IV Chaos Based Hardware Random Number Generator

The circuit is based on a chaos based hardware random number generator which
consists of an inductor and capacitance diode, called the Varactor.

Figure 1: Circuit Diagram

The figure 1 is a simple LCR circuit built around a varactor diode. Due to its
voltage dependent capacitance act as non linear element thus providing a
chaotic oscillation.

Generally a hardware random number generator is based on sampling noise
sources such as thermal noise or reverse based diode. Different circuits are
described in but these methods are difficult since there amplitudes are
usually small and often masked by deterministic disturbances so the other
alternative is to use a chaotic oscillator for pseudo random generator due to
its unpredictable behavior and relatively simple. We have built a chaotic
circuit based RNG (CCRNG), based on an earlier design by T.Kuusela kuusela ,
which in turn is built around the chaos generator of Matsumoto et. al
matsumoto . The nonlinear element here is the varactor, whose capacitance
varies as a function of voltage across it. The capacitance is varied as
$C(V)=C/(1+V/\theta)\gamma$, where V is the voltage across the diode. If the
circuit parameters and the external drive are suitably chosen, the system
exhibits period doubling and chaotic behaviour matsumoto .

Our circuit for chaotic clock Generator (CCG) is as shown in 1. It has an
external clock signal (A square wave of 500-600 KHz is used as a clock in this
case). Our CCG includes two identical chaotic oscillators so as to generate a
sequence of four bit structure in a random fashion, viz.,0001, 0010, 0100 and
1000. The heart of the unit is a pair of inductor 240$/mu$H and a varactor
diode BB609. An amplifier is used to raise the signal level so that if the
frequency and the amplitude of the driving clock is properly chosen the
circuit goes to chaotic state and a reliable operation is guaranteed even the
case of large tolerance. The fast operational amplifier, LF411, with a large
bandwidth is used as an amplifier. The voltage across the capacitor diode is a
random signal when the circuit exhibits period doubling and chaotic behaviour.
A Comparator (LM 311 with response time of 200ns) is used to convert analog
signal into a digital signal. Even though the output of the comparators are
quite random to generate it does not generate all the possible bit sequences.
To overcome this, we use two comparators for each CCRNG one of them takes bit
sample from the other this is done by a simple D-flip flop(1st D-flip flop).
To synchronize the output bit sequences with the driven clock one more D-flip
flop is used and a AND gate is used to avoid continues 1’s and 0’s. A decoder
is used to generate four bits structures using two CCRNG’s.

Using this circuit we generated several sets of random sequence of 0’s and 1’s
and tested its behaviour. At first we tapped the voltage at the output of the
chaotic circuit, at the edge of the varactor. This data stored onto the
computer as a function of time and its variation $\frac{dv}{dt}$ is computed.
Plotting $\frac{dv}{dt}$ against V gives a phase plot. Figure 2 shows behavior
of the generated signal. The plot on left side shows the raw data of voltage
v/s time while those on right side shows the phase plot $\frac{dV}{dt}$ v/s V.
As the frequency of the input clock circuit is changed the circuit goes from a
monotonic oscillator (figure A), to a bifurcation (figure C and D) and finally
to chaos (figure E). Output becomes chaotic when clock frequency is around 650
kHz (figure E).

Figure 2: Analog signal output of the Chaos Clock Generator at different
frequencies and corresponding Phase plots

The comparator converts this signal into a sequence of 0’s and 1’s. If the
output of chaotic circuit is above a set threshold value, the comparator gives
a one or else a zero. The output pulse is synchronized to the clock pulse. The
pulses are recorded on to the computer and further analyzed. these data are
obtained at the clock speed of about 650 kHz, since this is the region when
chaos circuit is giving a proper chaotic output.

Figure 3 shows two types of distribution. The total number of 1’s and 0’s are
shown in left side . This run is asymmetric since there is more 1’s than 0.
This happen only for a few runs and can easily be corrected by changing the
threshold value.

Figure 3: Histogram distribution of 0 and 1 (b) bit correlation

The graph on right, showing cross correlations is more important. It tells the
probability of getting zero following 1, getting a 1 after 1 and 0 after 0 and
1 after 0. the graph shows a slight higher value of occurrence of ‘1,1’
sequence but this is due to asymmetry of our circuit, which causes more
occurrences of 1’s as opposed to zero. More importantl, the graph clearly
shows same frequency for 10 and 01. This means that the system does not have
preference for 1 over 0, and a sequence of ’0,1’ is as probably as that of
’1,0’.

In addition, we studied the bit correlation of the data at different levels
that is $c_{ij}^{00}=<P_{i}(0)P_{j}(0)>$ where $P_{i}(0)$ and $P_{j}(0)$ are
the probability of finding 0 at $i^{th}$ position and $j^{th}$ position
respectively. Extending this calculation, we can write
$c_{ij}^{kl}=<P_{i}(k)P_{j}(l)>$ for k=1,0 and l=1,0. For an ideal case all
this should be equal and equal to a value 0.25. The graphs are shown in figure
4, where ‘Distance’ represents the gap between ‘$i$’ and ‘$j$’.

Figure 4: Successive bit correlation

However, figure 4 shows that $c_{ij}^{11}=0.35$ and $c_{ij}^{00}=0.9$ where as
$c_{ij}^{10}=c_{ij}^{01}=0.25$. This discrepancy in again due to the fact that
the circuit is slightly biased towards 1, due to the setting on the
discriminator. On the other hand the correlation between 01 and 10 are both
equal to 0.25, for all values of $i$ and $j$. indicating the circuit is a near
perfect coin toss system.

We also processed the sequence of 0’s and 1’s through the NIST test suite. The
P values for all the 15 tests are given in table 1

Statistical test | P-value
---|---
Frequency | 0.000199
BlockFrequency | 0.000003
CumulativeSums | 0.000439
CumulativeSums | 0.017912
Runs | 0.350485
LongestRun | 0.066882
Rank | 0
FFT | 0.017912
NonOverlappingTemplate | 0.004301
NonOverlappingTemplate | 0.002043
NonOverlappingTemplate | 0.350485
NonOverlappingTemplate | 0.035174
Table 1: RESULTS FOR THE UNIFORMITY OF P-VALUES AND THE PROPORTION OF PASSING
SEQUENCES

The P values are very low indicating that this is not a very reliable RNG,
even though the bit sequence is not correlated. This could be due to the
improper biasing of the system and needs to be further examined.

##  V Random number using LabVIEW

LabVIEW is a proprietary software by National Instruments Inc. USA, mainly
designed to interface hardware components of an experimental setup to the
computer. Since we extensively use this software to connect our data
acquisition and processing, we decided to test the in-built RNG or LabVIEW.
The in-built module of LabVIEW Produces a double precision, floating-point
number between 0 and 1, exclusively. Documentation of NI declares that this
module uses a multiple multiplicative congruential generators, using system
clock as a seed.

The output of this module has a uniform distribution. At first we test
$\frac{dn}{dt}$ v/s n, as shown in figure 5. In this case $n$ is the bit
output, with values $0$ or $1$. The figure essentially shows the probability
of getting a fixed value at time t and its dependence on previous value. The
highly non-periodic phase plot, indicates a randomness.

Figure 5: Phase plots

Figure 6: Histogram distribution of 0 and 1 (b) bit correlation

The figure 6 is histogram graph which is a visual representation of data that
measures the number of incidents of 0 and 1 of a sample set. left side shown
the distribution of 1’s and 0’s. We observe same frequency for 1’s and 0’s
which is true randomness behaviour. But the graph clearly shown same frequency
for 10 and 01. This means that the system does not have any performance for 1
over 0.

Figure 7: Successive bit correlation

As in case of earlier, we also compute correlation between $i^{th}$ bit and
$j^{th}$ bit. Except for a small bias, of about 0.3, for correlation for $i=1$
and $j=1$, remaining correlations, shown in figure 6 are nearly 0.25, for all
values of $i,j$.

We process the NIST suite for this data as well and get a better result. The
corresponding P values are shown in table 2.

Statistical test | P-value
---|---
Frequency | 0.739918
BlockFrequency | 0.350485
CumulativeSums | 0.534146
CumulativeSums | 0.534146
Runs | 0.911413
LongestRun | 0.739918
Rank | 0
FFT | 0.066882
NonOverlappingTemplate | 0.122325
NonOverlappingTemplate | 0.035174
NonOverlappingTemplate | 0.213309
NonOverlappingTemplate | 0.066882
NonOverlappingTemplate | 0.066882
NonOverlappingTemplate | 0.000954
NonOverlappingTemplate | 0.004301
NonOverlappingTemplate | 0.017912

Table 2: RESULTS FOR THE UNIFORMITY OF P-VALUES AND THE PROPORTION OF PASSING
SEQUENCES

These P values, shown in table 2 are more than 0.5 for almost all the tests.
It means its a good random that we can use for encryption.

##  VI Conclusion

Since software methods only offer a pseudo random number codes, need for other
sources is important. We have therefore analyzed hardware based random number
generator. In this we have compare and analyze the randomness behavior between
software and hardware random number generator. Although the correlation test
gives very good result, the NIST test shows it is yet a poor option, and that
needs to be rectified. The result of correlation indicates that this has a
promise and can be worked on. We show the in-built PRNG of LabVIEW shows a
relatively better performance. Additional work is needed to tweak our CCRNG to
show a better performance.

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11institutetext: Instituto de Ciencias Nucleares, Universidad Nacional
Autónoma de México, Circuito Exterior C.U., A.P. 70-543, México D.F. 04510,
México. 11email<EMAIL_ADDRESS>
# On Negative Mass Cosmology in General Relativity

Sebastián Nájera  Aldo Gamboa  Alejandro Aguilar-Nieto  Celia Escamilla-Rivera

In this Letter we present strong arguments in favour of thoroughly revising
the negative mass cosmology (NMC), which has been proposed as a simple
alternative explanation of dark energy and dark matter effects, within the
framework of General Relativity. We show that there are various physical
predictions of this model which require fine-tuning in order to make them
compatible with current cosmological surveys. In this way, the original
motivation of the NMC model becomes obscured due to the imposition of fine-
tuned unknown variables. We conclude that a more rigorous theoretical
treatment is needed in order to make the NMC a viable cosmological model.

###### Key Words.:

Cosmology – theory – dark energy – dark matter – observations – miscellaneous

Introduction.- A wide variety of observations currently favour the standard
model of cosmology $\Lambda$CDM, which one of its intrinsic theoretical
problems is the nature of the dark sector of the Universe. Along the years,
several proposals that span from theoretical and observational points of view
have treated successfully this issue, too many that to give fair references in
this introduction would be an extensive task. In this Letter we will focus our
discussion in one particular proposal: a physical model involving continuously
created negative masses in cosmology (Farnes, 2018), which we will denote as
NMC. This proposal is presented as an alternative explanation to the dark
sector of the Universe in a unified framework based on a Robertson-Walker (RW)
geometry.

The existence of negative masses is not a new idea (Bondi, 1957; Bonnor,
1989). Moreover, it naturally originates problems with theorems of energy
conditions (see e.g. Schon & Yau 1979; Witten 1981). Nevertheless, the
existence of physical phenomena which violate some of these theorems, led one
to question their validity (see e.g. Barcelo & Visser 2002 for a review on
violations of various energy conditions). Thus, despite its weirdness, the
proposal of negative masses has been used in several physical scenarios. For
example, it has been used to replace dark energy by a negative matter action
(Petit & d’Agostini, 2014), and, in a Schwarzschild space–time, it has been
found that there is no direct connection between the regions settled by the
negative and positive masses (Bondarenko, 2019).

Both positive and negative masses can coexist in the Universe under the
framework of GR, which assumes the validity of the weak equivalence principle.
This assumption naturally leads to the conclusion that positive mass particles
attract all massive particles, and negative mass particles repel all massive
particles (Bondi, 1957). However, the coexistence of negative and positive
masses triggers a peculiar phenomenon known as runaway motion, where two
particles of equal and opposite mass produce a constant acceleration of the
system towards the positive mass particle. This effect could have serious
implications in structure formation and the expansion of the Universe.
Therefore, we believe that a serious and rigorous analysis of this effect is
pertinent and should be performed.

Before starting this analysis, we should mention that the NMC model presented
by Farnes (2018) has received some strong critics, e.g. Socas-Navarro (2019)
identified several inconsistencies of this model with cosmological
observations, such as incorrect galactic halo masses, runaway galactic motions
or a different rate of cosmological structure formation. Also, Stepanian
(2019) made a heuristic analysis of the NMC model considering energy
conditions and observations from the early Universe, resulting in a mismatch.
However, our analysis discussed in this Letter goes beyond these ideas.

Negative Mass Cosmology.- In the NMC model the total density parameter can be
written as the contribution of each component of the Universe as

$\Omega=\Omega_{M+}+\Omega_{M-}+\Omega_{\Lambda},$ (1)

where $\Omega_{M+}$, $\Omega_{M-}$ and $\Omega_{\Lambda}$ are the critical
density parameters of positive mass, negative mass and cosmological constant,
respectively, and $\Omega_{k}=1-\Omega$ is the curvature critical density
parameter.

In this Universe we can have three different scenarios: (i) a positive-mass
dominated Universe, $|\Omega_{M+}|>|\Omega_{M-}|$, (ii) a massless cosmology,
$|\Omega_{M+}|=|\Omega_{M-}|$, in which there is an equal number of positive
and negative particles, and (iii) a negative-mass dominated Universe,
$|\Omega_{M+}|<|\Omega_{M-}|$, with special attention given to the latter
scenario as it deals with a possible cosmological constant effect produced by
the continuously created negative masses and thus there is no need for a
cosmological constant ($\Lambda=0$).

As it is standard in a RW space-time, the Friedmann equation is given
by111Throughout this Letter we will use geometrical units.

$H^{2}=\frac{\kappa}{3}(\rho_{+}+\rho_{-})+\frac{\Lambda}{3}-\frac{k}{a^{2}},$
(2)

where $H$ is the Hubble parameter, $\kappa=8\pi$, $a$ is the scale factor, $k$
is the curvature parameter, $\rho_{+}$ is the positive mass density and
$\rho_{-}$ is the negative mass density associated with a pressure $p_{-}$
through an equation of state $w_{-}=p_{-}/\rho_{-}$ for the negative masses.
Throughout this Letter we leave $w_{-}$ general, unless specifically stated.
According to the NMC model presented by Farnes (2018), in the case where
$\Lambda\neq 0$, there is a degeneracy between $\Omega_{M-}$ and
$\Omega_{\Lambda}$, which is given by
$\Omega_{\textrm{degen}}=\Omega_{M-}+\Omega_{\Lambda}$, and it is claimed that
in a conventional $\Lambda$CDM cosmology we are measuring this degeneracy.
Thus, by taking $\Omega_{M-}$ to be zero, we are falsely inferring a $\Lambda$
instead of a negative mass density parameter.

The degeneracy previously shown can be discriminated by the equation of state
$w_{-}$ of the negative masses, which yields $w_{-}=0$ for non-relativistic
matter. To resolve this degeneracy, the NMC model considers that negative
matter is constantly created by adding a matter creation term to the Einstein
field equations with $\Lambda$. This will result in an effective equation of
state for the negative mass fluid in which $w_{\textrm{eff}}\neq 0$. By adding
the creation term, the Einstein field equations,

$G_{ab}+\Lambda g_{ab}=\kappa T_{ab},$ (3)

where $G_{ab}$ is the Einstein’s tensor, $g_{ab}$ is the Robertson-Walker
metric, will be modified so that the energy-momentum tensor $T_{ab}$ is

$T_{ab}=(\rho+p+P_{c})u_{a}u_{b}+\left(p+P_{c}\right)g_{ab},$ (4)

with $\rho$ the total mass density, $p$ the total pressure, and

$P_{c}\coloneqq-\frac{\Gamma}{3H}(\rho_{-}+p_{-}),$ (5)

an effective pressure caused by the creation of negative mass particles, where
$\Gamma$ is the creation rate which, in principle, could depend on space
and/or time. The associated continuity equation for the negative masses is

$\dot{\rho}_{-}+3H(\rho_{-}+p_{-})=\Gamma(\rho_{-}+p_{-}).$ (6)

See e.g. Pan et al. (2016) for additional details of the background creation
of particles in cosmology. We note that the NMC model with $\Gamma=0$ refers
to the conventional negative mass cosmology within GR without creation of
negative mass particles.

The effective equation of state parameter for a Universe in which matter is
constantly created resembles that of a cosmological constant when $\Gamma=3H$.
Therefore, this NMC toy model has what appears to be an effective cosmological
constant $\Lambda_{-}\coloneqq 8\pi G\rho_{-}<0$. If we take
$\rho_{+}=\Lambda=0$ in this toy model, we can obtain the evolution of the
scale factor in a negative mass-dominated cosmology from Eq. (2). The solution
corresponds to an anti-de Sitter (AdS) Universe that undergoes a cycle of
expansion and contraction with a timescale of $\sqrt{-3\pi^{2}/\Lambda_{-}}$.
This cyclic cosmology does not match with our current observations, however
Farnes (2018) argued that there are two possible explanations for this: (i)
either the Universe is so large such that the local geometry appears to be
flat, or (ii) the presence of negative matter and its creation would modify
the CMB anisotropies. Both arguments would make the NMC toy model compatible
with early time observations. A few examples that employ this NMC toy model
are included in dark matter halos formation simulations, structure formation
simulations and also, in galaxy rotation curves, by means of cosmological
N-body simulations. Outside from this toy model, more research is needed in
order to understand the origin, the properties and the value of the creation
rate $\Gamma$.

Going a step forward and without assuming a specific form of $\Gamma$, we can
say more about this NMC model by analysing the deceleration parameter defined
by $q\coloneqq-\dot{H}/H^{2}-1$. Thus, using Eq. (2) the deceleration
parameter for this model is given by

$q=\frac{\Omega_{M+}}{2}+\frac{|\Omega_{M-}|}{2}\left[\left(\frac{\Gamma}{H}-3\right)(1+w_{-})+2\right]-\Omega_{\Lambda},$
(7)

which is independent of the curvature parameter $k$. Current observations
(Riess et al., 1998; Perlmutter et al., 1999) support that the Universe goes
through a late cosmic acceleration expansion, so we must have the correct
combination of variables in Eq. (7) to get $q(z=0)<0$. For the NMC toy model,
with $\Lambda=0$ and $\Gamma=3H$, Eq. (7) gives

$q=\frac{1}{2}\Omega_{M+}+|\Omega_{M-}|,$ (8)

which is independent of $w_{-}$. We notice in here that $q>0$ is preserved,
therefore the NMC toy model does not allow an accelerated expansion. This
result leads us to reject this toy model as a viable cosmological scenario,
since several observations apart from supernovae Type Ia as baryon acoustic
oscillations (Bassett & Hlozek, 2009), clusters of galaxies (Astier & Pain,
2012) and gravitational waves as standard sirens (Abbott et al., 2017) along
with the standard cosmography approach at low redshifts (Escamilla-Rivera &
Capozziello, 2019) confirm this cosmic acceleration.

Moreover, it is always possible to choose different values for $\Gamma$,
$\Omega_{\Lambda}$ and even $w_{-}$ (if we try to give a more exotic nature to
the negative masses), to get the required accelerated expansion. Nevertheless,
the choices for these values might be non-physical and they would
significantly increase the complexity of the problem and would erase the
initial motivation of a NMC model as a simple alternative explanation of the
dark sector.

Following the above general ideas, a final comment can be made for a negative-
mass dominated cosmology (i.e. $|\rho_{-}|\gg\rho_{+}$) with $\Lambda=0$. From
Eq. (1), these assumptions imply $k=-1$ and thus it is not always possible to
have physical solutions compatible with observations (see comments below),
since $H^{2}$ could take negative values. In particular, in a negative dust
Universe without creation of particles ($w_{-}=0$, $\Gamma=0$), $H^{2}$
changes sign in

$a=\frac{|\Omega_{-,0}|}{\Omega_{k,0}},$ (9)

with $\Omega_{i,0}=\Omega_{i}|_{t=0}$, where the subindex $i$ denotes each
fluid component. The same pathology occurs in the case $\Lambda>0$, where once
again $H^{2}$ could become negative for a value of $a>0$. By similar
arguments, $H^{2}$ could still be negative when considering radiation and the
creation of negative mass particles, because $\rho_{-}<0$ independently of
$w_{-}$ and $\Gamma$, as can be shown from Eq. (6). Therefore, within the NMC
model we have a Universe which could begin with a finite size. Nevertheless,
we know from estimations of the early Universe in the standard $\Lambda$CDM
model, that the Universe must begin from a very compact and dense state
(Weinberg, 2008; Ellis et al., 2012) to be able to predict cosmological
observables, such as the abundances of primordial nuclei (Cyburt et al., 2016;
Steigman, 2007; Iocco et al., 2009) and other early universe observations
(Aghanim et al., 2020; Cruz & Escamilla-Rivera, 2021). In this way, we must
fine-tune the nature of $\rho_{-}$ and $\Gamma$ in order to make the NMC scale
factor compatible with these observables.

Runaway motion.- Considering comoving coordinates and a smooth one-parameter
family of geodesics $\gamma_{s}(t)$, we denote the unit timelike tangent
vector field to the family of geodesics as $\mathcal{T}^{a}=(\partial/\partial
t)^{a}$, such that $\mathcal{T}^{a}\mathcal{T}_{a}=-1$, and the separation
vector between nearby geodesics as $X^{a}=(\partial/\partial r)^{a}$, which
satisfies the orthogonality property $\mathcal{T}_{a}X^{a}=0$. In this way,
the geodesic deviation equation is

$A^{a}=-\tensor{R}{{}_{c}{}_{b}{}_{d}^{a}}X^{b}\mathcal{T}^{c}\mathcal{T}^{d},$
(10)

where
$A^{a}=\mathcal{T}^{c}\nabla_{c}v^{a}=\mathcal{T}^{c}\nabla_{c}\left(\mathcal{T}^{b}\nabla_{b}X^{a}\right)$
is the relative acceleration between nearby geodesics and
$\tensor{R}{{}^{a}_{b}{}_{c}{}_{d}}$ is the Riemann tensor. In particular, it
is well known that for fundamental observers in RW space–times,
$A^{a}=-qH^{2}(\partial/\partial r)^{a}$. Since $q=q(t)$ and $H=H(t)$, the
relative acceleration must be common to all observers, and due to the
symmetries of the Robertson–Walker metric we must deal with other geometries
in order to estimate the effects of runaway motion.

We now consider the particular case of negative and positive particles in an
arbitrary space–time. Using the expansion of the Riemann tensor

$R_{abcd}=C_{abcd}+\left(g_{a[c}R_{d]b}-g_{b[c}R_{d]a}\right)-\frac{1}{3}Rg_{a[c}g_{d]b},$
(11)

where $C_{abcd}$ is the Weyl tensor, and $R_{ab}$ is the Ricci tensor we can
rewrite Eq. (10) as

$\displaystyle A^{a}$ $\displaystyle=$
$\displaystyle-\left[C_{cbde}+\left(g_{c[d}R_{e]b}-g_{b[d}R_{e]c}\right)-\frac{1}{3}Rg_{c[d}g_{e]b}\right]$
(12) $\displaystyle g^{ea}X^{b}\mathcal{T}^{c}\mathcal{T}^{d}.$

Since the Weyl tensor is related to free space and the Ricci scalar is related
to the energy–momentum tensor through the Einstein field equations, we can
divide the geodesic deviation equation into an acceleration due to free space
$A^{a}_{\text{FS}}$, and the acceleration due to matter $A^{a}_{\text{M}}$,
defined by

$\displaystyle A^{a}_{\text{FS}}$ $\displaystyle\coloneqq$
$\displaystyle-\tensor{C}{{}_{c}{}_{b}{}_{d}^{a}}X^{b}\mathcal{T}^{c}\mathcal{T}^{d},$
(13) $\displaystyle A^{a}_{\text{M}}$ $\displaystyle\coloneqq$
$\displaystyle-\left[\left(g_{c[d}R_{e]b}-g_{b[d}R_{e]c}\right)-\frac{1}{3}Rg_{c[d}g_{e]b}\right]$
(14) $\displaystyle\,g^{ea}X^{b}\mathcal{T}^{c}\mathcal{T}^{d}.$

If we consider that the acceleration is due solely to matter and use the
Einstein field equations $R_{ab}=\kappa(T_{ab}-\frac{1}{2}Tg_{ab})$ and
$R=-\kappa T$, then Eq. (14) reads as

$A^{a}_{\text{M}}=\frac{\kappa}{2}\left[\tensor{T}{{}^{a}_{b}}X^{b}-\mathcal{Q}_{b}X^{b}\mathcal{T}^{a}-\left(\frac{\rho}{3}+2p\right)X^{a}\right],$
(15)

where we have used that $T=g^{ab}T_{ab}=-\rho+3p$, with the energy density
$\rho=u^{a}u^{b}T_{ab}$, the isotropic pressure $p=(g^{ab}+u^{a}u^{b})T_{ab}$
and the definition of the spatial energy flux vector
$\mathcal{Q}_{a}\coloneqq-\mathcal{T}^{b}T_{ab}$.

So far, the matter content is completely general. We will restrict to a
special case. We consider the matter content as two compact fluids, the first
composed of matter with positive density and the second with negative density.
These fluids are separated such that we can neglect gravitational effects due
to the opposite fluid and thus we can consider both as comoving. With these
assumptions, we can rewrite the energy-momentum tensor as

$T_{ab}=(\rho_{+}-|\rho_{-}|)\mathcal{T}_{a}\mathcal{T}_{b},$ (16)

and from Eq. (15), we get

$A^{a}_{\text{M}}=\frac{\kappa}{6}\left[-\rho_{+}+|\rho_{-}|\right]X^{a}.$
(17)

These results can be corroborated with Ellis & Van Elst (1999) for positive
masses and a FRW geometry. Thus, observers nearby to the positive particles
will be attracted while observers in a neighbourhood of the negative particles
will be repelled. Physically, this represents the so-called runaway motion.
Moreover, using the NMC’s effective energy–momentum tensor in Eq. (4),
$A^{a}_{\text{M}}$ reads as

$A^{a}_{\text{M}}=\frac{\kappa}{6}\left\\{-\rho_{+}+|\rho_{-}|\left[\frac{\Gamma}{H}(1+w_{-})+3w_{-}+1\right]\right\\}X^{a},$
(18)

and considering $w_{-}=0$ the latter equation can be reduced to

$A^{a}_{\text{M}}=\frac{\kappa}{6}\left[-\rho_{+}+|\rho_{-}|\left(\frac{\Gamma}{H}+1\right)\right]X^{a},$
(19)

therefore, we obtain analogous results to the dust case. We notice that,
unlike the fundamental observers in RW space–times, the relative acceleration
in Eqs. (17)-(19), is not homogeneous nor isotropic, hence in a NMC model with
$|\rho_{-}|\left(\frac{\Gamma}{H}+1\right)>|\rho_{+}|$, we have a positive
relative acceleration between nearby geodesics, which could cause non-linear
effects in the velocity perturbations. Thus, structure formation in this kind
of cosmology becomes a fine-tuning problem.

Concluding remarks.- In this letter we thoroughly analysed a negative mass
cosmological (NMC) model which arise as an alternative, simpler and unified
explanation of the dark sector of the Universe, within the context of General
Relativity, and with a possible background of continuously created negative
mass particles. We found that this model requires fine-tuning in order to
predict a late cosmological accelerated expansion, to have a Hubble parameter
defined in the whole cosmic redshift range expected from early Universe
observations, and to allow structure formation. Moreover, we discarded the
reviewed model with a negative mass creation rate of $\Gamma=3H$, because it
does not predict late accelerated expansion; this particular choice of
$\Gamma$ had been used to explain and numerically model the effects of dark
energy and dark matter. In its current state, the NMC model requires a more
concise theoretical formulation before being considered as a viable model of
the Universe which can be tested with cosmological surveys.

Acknowledgments.- AA, AG and SN acknowledge financial support from CONACYT
postgraduate grants program. CE-R acknowledges the Royal Astronomical Society
as FRAS 10147 and by DGAPA-PAPIIT-UNAM Project IA100220. This article is also
based upon work from COST action CA18108, supported by COST (European
Cooperation in Science and Technology). The ideas treated in this Letter were
derived from a discussion in the Lecture entitled “Aplicaciones Astrofísicas y
Cosmológicas de la Relatividad General” at ICN-UNAM.

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