Is Dark Matter Trapped Dynamic-Density Dark Energy Or Ancient CMB?

 by Steven Gussman

        The mechanism behind the present accelerated metric expansion of the universe is not yet well-understood.  Most often, astrophysicists deduce that there must be a "dark" energy of a particular density associated with this expansion.  This is not particularly strange until you realize that the claim is that it is a static scalar field (that is, a field whose density does not depend on its volume; shrink or stretch space and the density of dark energy remains constant).  This rather obviously violates the conservation of energy (of course dark energy must indeed be created and destroyed when spacetime is expanded or contracted).  Astrophysicist Ethan Siegel is forthright about this: he believes, due to the empirical expansion of the universe, that energy is not strictly conserved (and he admirably wants the mainstream to admit that this is their necessary belief as well).[1]

        I must confess that this never made much sense to me, and is another of the many claims of "modern physics" which seem half-baked.[2]  But can we explain dark matter by taking the strangeness of dark energy (partially) seriously?  The astrophysicists believe they can hold the contradictory statement that dark energy is a genuinely scalar field, but also that it does not violate the conservation of energy.  For our purposes, let's take their dark energy field more seriously than they do: let us assume that more dark energy is created (somehow) as spacetime expands, maintaining its constant density.  Let us further assume that it is this dynamical growth rate (which matches the space-time expansion rate to maintain the density) that is the inexorable fact about the dark energy field, and not the constant density per se.

        It is explained that the space between myself and my laptop (and in fact between the stars of a galaxy, and even mutual galaxies in a group) is not expanding because the gravitational attraction of these matter-dense loci overpowers dark energy (which is really quite weak except in the vicinity of empty space).[3]  Interestingly, about 90% of the galactic mass that allegedly puts the breaks on dark energy's spacetime expansion is not stellar, but in fact another mystery--dark matter[4] (though locally, it is thought that the ordinary gravity of our own sun is enough to stop the expansion of space in the solar system).

        Interestingly, if we assume, as mentioned before, that dark energy is somehow replicating itself, then this should not cease to happen in the vicinity of a gravitational field such as that inside of a galaxy (only the expansion of that spacetime need stop).  Here we are presented with an alternative picture in which, inside of gravitational wells, dark energy fails to expand space, but still grows in energy.  In this model, inside of gravitational wells, the dark energy density grows (it still does not do so out in the emptiest regions of space--the vast majority of space--and instead expands spacetime, there).  Could it be that galaxies form with ordinary matter / gravity, stop the local expansion of space, and then exponentially increase in mass over time due to the trapped, growing dark energy inside of them, thereby accounting for dark matter (both dark matter and dark energy get their names from the fact that they are indirectly deduced fields of energy which are understood to be spread out and non-interacting with light)?

        I will attempt a back-of-the-envelope calculation for our own Milky Way Galaxy.  As initial conditions, we will take the spherical volume with the radius of our galaxy; we will take the ordinary dark energy density to be its natural starting point; and we will take the growth rate of dark energy.  The radius of the observable universe, due to post-inflationary expansion, has increased one-thousand-fold (as the temperature of the CMB was red-shifted from ~3,000 K to ~3 K).  In terms of volume, this means that the observable universe has grown by a factor of (10³)³ = 10⁹ (one-billion-fold).  Out in empty space, we believe that this expansion was due to the field of dark energy keeping pace in its own growth to maintain its scalar density.  Within a galaxy, we will assume then that the energy (and therefore density) of local dark energy increased by a factor of 10⁹.  The question is: does a constant-volume of such a field of replicating dark energy, beginning with the density of dark energy in empty space, and growing by a factor of 10⁹ over the following 13.8 billion years, equal the dark matter mass of said galaxy?

        Given:

        c = 2.99792458 × 10⁸ m/s[5]
        M_☉ = (1.98847 ± 0.00007) × 10³⁰ kg[6]
        p[Dark Energy] = 2.2 × 10^-27 kg/m³[7]
        p[CMB] = 4.64 × 10^-31 kg/m³[8]
        M[Milky Way] = 2.29 × 10⁴² kg[9]
        M[Milky Way Dark Matter] = 2 × 10⁴² kg[4]
        r[Milky Way] = (4.13 ± 0.170) × 10²⁰ m[10]
        h[Milky Way] = 4.5 × 10¹⁹ m (6.8 × 10¹⁸ m - 1.5 × 10¹⁹ m)[11]

        We will begin by calculating the volume of the Milky Way.  This is a bit of a troublesome approximation, as our galaxy is a disk (not a sphere), though gravity itself does indeed bind radially (and in fact there is a low-density spherical region called the stellar halo).[12]

        V[Sphere] = (4/3)πr³
        V[Milky Way] = (4/3)πr[Milky Way]³
        V[Milky Way] = (4/3)π(4.13 × 10²⁰ m)³
        V[Milky Way] = 2.95 × 10⁶² m³

        Next, we calculate the late-density of the supposedly trapped dark energy:

        p[Trapped Dark Energy] = p[Dark Energy] × 10⁹
        p[Trapped Dark Energy] = 2.2 × 10^-27 kg/m³ × 10⁹
        p[Trapped Dark Energy] = 2.2 × 10^-18 kg/m³

        Finally, we calculate the total mass of this trapped dark energy:

        M = pV
        M[Trapped Dark Energy] = p[Trapped Dark Energy] × V[Milky Way]
        M[Trapped Dark Energy] = (2.2 × 10^-18 kg/m³) × (2.95 × 10⁶² m³)
        M[Trapped Dark Energy] = 6.5 × 10⁴⁴ kg

        Now because this is meant to be a dark matter candidate, we compare it to the mass of dark matter in the Milky Way galaxy:

        M[Trapped Dark Energy] / M[Milky Way Dark Matter]
        = (6.5 × 10⁴⁴ kg) / (2 × 10⁴² kg)
        = 300

        This model predicts at least two orders of magnitude more mass than what is known to exist in the Milky Way, meaning it is untenable.  But perhaps the Milky Way's volume needs to be treated as a cylinder due to the fact that it is a rather thin disk.

        V = πr²h
        V[Milky Way]' = πr[Milky Way]²h[Milky Way]
        V[Milky Way]' = π(4.13 × 10²⁰ m)²(4.5 × 10¹⁹ m)
        V[Milky Way]' = 2.4 × 10⁶¹ m³

        M[Trapped Dark Energy]' = p[Trapped Dark Energy]V[MilkyWay]'
        M[Trapped Dark Energy]' = (2.2 × 10^-18 kg/m³)(2.4 × 10⁶¹ m³)
        M[Trapped Dark Energy]' = 5.3 × 10⁴³ kg

        M[Trapped Dark Energy]' / M[Milky Way Dark Matter]
        = (5.3 × 10⁴³ kg) / (2 × 10⁴² kg)
        = 30

        Here we obtain an answer that is only one order-of-magnitude too large.  If we use the lower end of the height-estimate for the Milky Way's stellar disk, we could probably obtain an order-of-magnitude match (indeed, we get 8E42 kg--only about four times too large).  Yet as it is, the dark matter halo is thought to be a significantly larger spherical region than the visible stellar-disk-as-cylinder, which would increase the estimate in the wrong direction.  This suggests that the toy model probably does not work.

        However, there is another possibility.  As stated early, in the standard model, the CMB's energy is red-shifted from ~3,000 K to ~3 K.  As space expanded, not only was the wavelength stretched thinner, but the photon-density was spread out.  This suggests the photon-density of the CMB was likewise reduced by a factor of 1,000³ = 10⁹.  This suggests the total energy density of the CMB was reduced by a factor of 10³ (by red-shifting) × 10⁹ (by reduced photon density) = 10¹².[13]  Yet, again, in the region of galaxies, space did not expand, and so the CMB in these regions should be significantly more ancient (higher energy) in nature.  Now, what happens if we pretend the CMB in our neighborhood did not in fact cool (a possibility with serious empirical issues)?

        p[Early CMB] = p[CMB] × 10¹²
        p[Early CMB] = 4.64 × 10^-31 kg/m³ × 10¹²
        p[Early CMB] = 4.64 × 10^-19 kg/m³

        M[Trapped CMB] = p[Early CMB] × V[Milky Way]
        M[Trapped CMB] = (4.64 × 10^-19 kg/m³) × (2.95 × 10⁶² m³)
        M[Trapped CMB] = 1.37 × 10⁴⁴ kg

        M[Trapped CMB] / M[Milky Way Dark Matter]

        = (1.37 × 10⁴⁴ kg) / (2 × 10⁴² kg)
        = 70

        Suggesting that if this were the case, it would be an order-of-magnitude too large (and again, potentially recoverable by the use of a more accurate volume, deviating from a sphere towards a cylinder).  This is an issue as well because such light moves as fast as possible in all directions.  Even if The Milky Way existed in more-or-less its present form (in terms of stellar matter) since the beginning of time, CMB which was and wasn't being affected by the expansion of the universe would be passing in and (mostly) out of the galaxy's gravity well all of the time.  It is nice to think that perhaps this could allow the creation of "dark matter" (heavy CMB) towards the galaxy's center, which moved outward creating a "dark matter halo" (with light CMB having moved in near our sun) over time, but this would require a far more careful, dynamic analysis.

        These models currently have too much in the way of free parameters (what is taken to be the volume, for example), and their detailed dynamics might be very important compared to these back-of-the-envelope calculations.  The back-of-the-envelope calculations attempted here do not inspire great confidence that such mechanisms could explain dark matter, though they are ultimately inconclusive due to their lack of precision and inability to capture the full dynamics predicted by such models.

Footnotes:

1. See "Strange But True: The Expanding Universe Doesn’t Conserve Energy" by Ethan Siegel (Big Think) (2023) (https://bigthink.com/starts-with-a-bang/expanding-universe-conserve-energy/). I have posited a potential model in which the field is only approximately scalar and in which energy is conserved at the level of a multiverse, see "Towards A Falsifiable Black-Hole Cosmos" by Steven Gussman (Footnote Physicist) (2023) (https://footnotephysicist.blogspot.com/2023/02/towards-falsifiable-black-hole-cosmos.html).

2. It is fine for astrophysical cosmology knowledge to be in the state that it is in--in fact, it's miraculous how much we have learned so quickly. I simply believe a dose of humility is in order as well. Why don't we say, "it's as if there is a scalar field which violates the conservation of energy permeating the universe, so there must be something we don't understand, yet?" The mainstream alternative, to simply say that we know such a field is at play, and that it actually is scalar, but also doesn't violate the laws of thermodynamics, is untenable (you cannot have your cake and eat it too--how are the energy debts of a scalar field being paid? Do you believe they have to be paid? And so on).

3. See "This Is Why We Aren't Expanding, Even If The Universe Is" by Ethan Siegel (Forbes) (2019) (https://www.forbes.com/sites/startswithabang/2019/02/19/this-is-why-we-arent-expanding-even-if-the-universe-is/?sh=5c979b465311)† and "Local Group" (Wikipedia) (accessed 8/2023) (https://en.wikipedia.org/wiki/Local_Group).†

4. See "Milky Way" which further cites "The Latest Calculation Of Milky Way's Mass Just Changed What We Know About Our Galaxy" by Starr and "Evidence For An Intermediate-Mass Milky Way From Gaia DR2 Halo Globular Cluster Motions" by Watkins et al.

5. See "Speed Of Light" (Wikipedia) (accessed 8/4/23) (https://en.wikipedia.org/wiki/Speed_of_light).† (Note: † denotes that I have no read a piece in its entirety, and †† denotes that I have not read it at all).

6. See "Solar Mass" (Wikipedia) (accessed 8/4/23) (https://en.wikipedia.org/wiki/Solar_mass)† which further cites "Astronomical Constants" (The Astronomical Almanac) (2013) (https://web.archive.org/web/20131110215339/http://asa.usno.navy.mil/static/files/2014/Astronomical_Constants_2014.pdf).††

7. See "Dark Energy" (Wikipedia) (accessed 8/4/23) (https://en.wikipedia.org/wiki/Dark_energy)† which further cites "Why The Cosmological Constant Is Small And Positive" by Paul J. Steinhardt and Neil Turok (Science) (2006) (https://arxiv.org/abs/astro-ph/0605173)††, "Dark Energy" (Hyperphysics) (2013) (https://web.archive.org/web/20130527105518/http://hyperphysics.phy-astr.gsu.edu/HBASE/astro/dareng.html)††, and "Dark Matter(Dark Energy)" by Timothy Ferris (2015) (https://web.archive.org/web/20150610172523/http://ngm.nationalgeographic.com/2015/01/hidden-cosmos/ferris-text)††; and "Dark Energy" by R. Nave (HyperPhysics) (http://hyperphysics.phy-astr.gsu.edu/hbase/Astro/dareng.html).††

8. See "Cosmic Microwave Background" (Wikipedia) (accessed 8/4/2023) (https://en.wikipedia.org/wiki/Cosmic_microwave_background)† which further cites "29. Cosmic Microwave Background" by P.A. Zyla et al. (LBL, Berkeley / Particle Data Group) (https://pdg.lbl.gov/2020/reviews/rpp2020-rev-cosmic-microwave-background.pdf).††

9. See "Milky Way" (Wikipedia) (accessed 8/4/23) (https://en.wikipedia.org/wiki/Milky_Way)† which further cites "The Latest Calculation Of Milky Way's Mass Just Changed What We Know About Our Galaxy" by Michelle Starr (ScienceAlert) (2019) (https://web.archive.org/web/20190308125656/https://www.sciencealert.com/the-most-accurate-measurement-yet-of-the-milky-way-s-mass-puts-us-ahead-of-andromeda)††, "Evidence For An Intermediate-Mass Milky Way From Gaia DR2 Halo Globular Cluster Motions" by Laura L. Watkins et al. (The Astrophysical Journal) (2019) (https://arxiv.org/abs/1804.11348)††, and "Kinematics Of The Stellar Halo And The Mass Distribution Of The Milky Way Using Blue Horizontal Branch Stars" by P.R. Kafle, S. Sharma, G. F. Lewis, J. Bland-Hawthorn (The Astrophysical Journal) (2012) (https://arxiv.org/abs/1210.7527).††

10. See "Milky Way" which further cites "The Milky Way Is Just An Average Spiral" by S. P. Goodwin, J. Gribbin, and M. A. Hendry (arXiv) (1997) (https://arxiv.org/abs/astro-ph/9704216)††, "The Relative Size Of The Milky Way" by S. P. Goodwin, J. Gribbin, and M. A. Hendry (The Observatory) (1998) (https://ui.adsabs.harvard.edu/abs/1998Obs...118..201G)††, and "Warps And Correlations With Intrinsic Parameters Of Galaxies In The Visible And Radio" by N. Castro-Rodríguez, M. López-Corredoira, M. L. Sánchez-Saavedra, and E. Battaner (Astronomy & Astrophysics) (2002) (https://arxiv.org/abs/astro-ph/0205553).††

11. See "Milky Way" which further cites "The Galaxy In Context: Structural, Kinematic, And Integrated Properties" by Joss Bland-Hawthorn and Ortwin Gerhard (Annual Review of Astronomy and Astrophysics) (2016) (https://arxiv.org/abs/1602.07702).††

12. See "Stellar Halo" (Wikipedia) (accessed 8/4/23) (https://en.wikipedia.org/wiki/Stellar_halo)† which further cites "The Structure Of The Galactic Halo" by F. D. A. Hartwick (Proceedings Of The NATO Advanced Study Institute) (1987) (https://ui.adsabs.harvard.edu/abs/1987ASIC..207..281H/abstract).††

13. I think that this reasoning is roughly corroborated by looking over most of "Lecture 31: The Cosmic Microwave Background Radiation" by Dmitri Pogosyan (University Of Alberta) (https://sites.ualberta.ca/~pogosyan/teaching/ASTRO_122/lect31/lecture31.html).†