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European Journal of Applied Sciences – Vol. 11, No. 2

Publication Date: April 25, 2023

DOI:10.14738/aivp.112.14383. Hoeneisen, B. (2023). A Data Driven Solution to the Dark Matter Problem. European Journal of Applied Sciences, Vol - 11(2). 473-

481.

Services for Science and Education – United Kingdom

A Data Driven Solution to the Dark MatterProblem

Bruce Hoeneisen

Universidad San Francisco de Quito, Quito, Ecuador

ABSTRACT

A data driven solution to the dark matter problem is presented. This short and self- contained overview 1is intended for a wide audience, with full technical details

available in the cited references. We present redundant, independent and

consistent measurements of the dark matter particle comoving root-mean-square

velocity vhrms(1), or equivalently, of the dark matter temperature-to-mass ratio.

These measurements agree with the “no freeze-in and no freeze-out” scenario of

spin zero dark matter that decouples early on from the Standard Model sector, e.g.,

spin zero dark matter coupled to the Higgs boson or to the top quark. 1

Keywords: Warm Dark Matter, Galaxy Rotation Curves, Galaxy UV Luminosity, Dwarf

Galaxies

INTRODUCTION

Most of the matter in the universe, 84.3 ± 0.2 % [1], is in a “dark matter”form that has

been “observed” only through its gravitational interaction. As far as we know, this dark matter

does not have any other interaction, at least down to the current sensitivity of our experiments

and observations. Fritz Zwicky in 1933 found that the matter in the Coma cluster of galaxies

greatly exceeds the mass in stars [2]. According to the book by Stefano Profumo [3], the mass

mh of dark matter particles is unknown over 90 orders of magnitude! In summary, we know

exactly how much dark matter there is, but do not have the foggiest idea what it is. This

article presents an overview of a data driven solution to the dark matter problem.

MEASUREMENTS

Let us assume that dark matter is a gas of particles that is ultra-relativistic in the early universe.

As the universe expands, dark matter cools, and becomes non-relativistic. We expect that this

non-relativistic gas is classical, i.e., has negative chemical potential [4] [5]. Let vhrms(a) be the

root-mean-square velocity of the non-relativistic dark matter particles. a(t) is the expansion

parameter of the universe at time t, normalized to a(t0) = 1 at the present time. 2 v2hrms(a) is

proportional to the dark matter temperature-to-mass ratio. As the universe expands, dark

matter cools, vhrms(a) varies in proportion to 1/a [6], and the dark matter density ρh(a) varies

in proportion to 1/a3, so

1 Presented at the 4th World Summit on Exploring the Dark Side of the Universe, La R ́eunion, November 7-11

2022

2 To understand the expansion parameter a(t), imagine a baloon covered with dots. As the baloon is inflated, the

distances between neighboring dots, i.e., galaxies, increase in proportion to a(t).

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European Journal of Applied Sciences (EJAS) Vol. 11, Issue 2, April-2023

does not depend on a. In other words, we say that vhrms(1) is an “adiabatic invariant”.

It turns out that, to unravel the dark matter mystery, we need to measure the adiabatic

invariant vhrms(1), and the related observable kfs ≡ 2π/λfs that we now explain. Due to the

velocity dispersion vhrms(a), dark matter particles free-stream in and out of density minimums

and maximums, erasing primordial density fluctuations of “comoving” wavelength less than

approximately λfs [4] [7]. (Since wavelengths grow in proportion to a(t), it is customary to refer

the wavelength to the present time, i.e., a(t0) = 1, hence the word “comoving”.) kfs is the

comoving cut-off wavenumber due to dark matter free-streaming. The theoretical relation

between the two observables, vhrms(1) and kfs, is summarized in Table 1 [4] [7].

Figure 1

Figure 1: Shown are distributions of x, where x is the observed galaxy stellar mass M∗/M⊙

times 101.5 (stars) [8] [9] [10], or the observed galaxy UV luminosity νLUV/L⊙ (squares)

[11] [12] [13] (corrected for dust extinction [14] [15]), or the predicted linear total (dark

matter plus baryon) mass M/M⊙ (lines), at redshift z = 6. The symbol ⊙ means “sun”. The

Press- Schechter prediction, and its ShethTormen ellipsoidal collapse extensions,

correspond, from top to bottom, to the warm dark matter free-streaming cutoff

wavenumbers kfs = 1000, 4, 2 and 1 Mpc−1. The round red, blue and green dots indicate the

velocity dispersion cut-offs of the predictions [16] atkfs = 1, 2 and 4 Mpc−1, respectively.

Presenting three predictions illustrates the uncertainty of the predictions. Note that the

data agree with predictionsfor kfs ≈ 2 Mpc−1.

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Hoeneisen, B. (2023). A Data Driven Solution to the Dark Matter Problem. European Journal of Applied Sciences, Vol - 11(2). 473-481.

URL: http://dx.doi.org/10.14738/aivp.112.14383.

Table 1: Calculated relation between the adiabatic invariant vhrms(1), and the

comoving cut-off wavenumber kfs, due to dark matter free-streaming [4][7].

“Mega-parsec” (Mpc) is a unit of length used in cosmology.

To measure kfs we compare observed and predicted galaxy rest-frame ultraviolet luminosity

distributions, and observed and predicted galaxy stellar mass distributions. An example,

corresponding to “redshift” z = 6, or equivalently, expansion parameter a = 1/(1 + z) = 1/7,

is shown in Figure 1. Note that warm dark matter free-streaming attenuates small scale

fluctuations and therefore reduces the numbers of galaxies with low mass. From this, and

similar figures at redshifts z = 4, 8 and 10, we obtain [17].

To measure the adiabatic invariant vhrms(1) we note the following. Consider a free observer in

a density peak in the early universe. This observer “sees” dark matter expand adiabatically, i.e.

conserving vhrms(1), due to the expansion of the universe, reach maximum expansion, followed

by adiabatic compression into the core of the galaxy due to gravitational attraction. The core of

the galaxy forms adiabatically if dark matter is warm as we have assumed, i.e if vhrms(1) is

greater than zero [18]. Rotation and relaxation, due to galaxy collisions and mergers, increase

the observed v′hrms(1) above the true vhrms(1) [6]. So, as long as rotation and relaxation remain

negligible, we predict that the adiabatic invariant in the core of the galaxy is the same as in the

early universe, and so should be the same for all galaxies (with negligible rotation and

relaxation). The adiabatic invariant in the core of a spiral galaxy can be obtained from the

observed rotation curves of neutral atomic hydrogen gas or of stars, together with infrared and

visible images [19]. The distribution of the adiabatic invariant measured in several dwarf

galaxies is shown in Figure 2. We obtain a narrow peak with vhrms(1) = 406 ± 69 m/s [6]. The

few galaxies with v′hrms(1) to the right of this peak have significant rotation and/or relaxation.

We note, from Table 1, that the measurements of kfs and vhrms(1) are consistent with each other.

We conclude that, 1) vhrms(1) in the core of galaxies (corrected for dark matter rotation and

relaxation) is of cosmological origin, as inferred form the narrowness of the peak in Figure 2,

and as predicted for warm dark matter [18][21]; and 2) that kfs is indeed due to warm dark

matter particle free-streaming.