Lately I have been seeing numerous articles on right hand neutrinos contributing to dark matter. There are several different proposals. Those proposals involve whether or not neutrinos follow the terms of Dirac mass or Majorana mass

https://arxiv.org/abs/2008.02110

here is a breakdown into singlets and doublets

SU(2)

\[{\small\begin{array}{|c|c|c|c|c|c|c|c|c|c|}\hline Field & \ell_L& \ell_R &v_L&U_L&d_L&U_R &D_R&\phi^+&\phi^0\\\hline T_3&- \frac{1}{2}&0&\frac{1}{2}&\frac{1}{2}&-\frac{1}{2}&0&0&\frac{1}{2}&-\frac{1}{2} \\\hline Y&-\frac{1}{2}&-1&-\frac{1}{2}&\frac{1}{6}&\frac{1}{6}& \frac{2}{3}&-\frac{1}{3}&\frac{1}{2}&\frac{1}{2}\\\hline Q&-1&-1&0&\frac{2}{3}&-\frac{1}{3}&\frac{2}{3}&-\frac{1}{3}&1&0\\\hline\end{array}}\]

\(\psi_L\) doublet

\[D_\mu\psi_L=[\partial_\mu-i\frac{g}{\sqrt{2}}(\tau^+W_\mu^+\tau^-W_\mu^-)-i\frac{g}{2}\tau^3W^3_\mu+i\acute{g}YB_\mu]\psi_L=\]\[\partial_\mu-i\frac{g}{\sqrt{2}}(\tau^+W_\mu^-)+ieQA_\mu-i\frac{g}{cos\theta_W}(\frac{t_3}{2}-Qsin^2\theta_W)Z_\mu]\psi_L\]

\(\psi_R\) singlet

\[D_\mu\psi_R=[\partial\mu+i\acute{g}YB_\mu]\psi_R=\partial_\mu+ieQA_\mu+i\frac{g}{cos\theta_W}Qsin^2\theta_WZ_\mu]\psi_W\]

 with \[\tau\pm=i\frac{\tau_1\pm\tau_2}{2}\] and charge operator defined as

\[Q=\begin{pmatrix}\frac{1}{2}+Y&0\\0&-\frac{1}{2}+Y\end{pmatrix}\]

\[e=g.sin\theta_W=g.cos\theta_W\]

\[W_\mu\pm=\frac{W^1_\mu\pm iW_\mu^2}{\sqrt{2}}\]

\[V_{ckm}=V^\dagger_{\mu L} V_{dL}\]

The gauge group of electroweak interactions is 

\[SU(2)_L\otimes U(1)_Y\] where left handed quarks are in doublets of \[ SU(2)_L\] while right handed quarks are in singlets

the electroweak interaction is given by the Langrangian

\[\mathcal{L}=-\frac{1}{4}W^a_{\mu\nu}W^{\mu\nu}_a-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}+\overline{\Psi}i\gamma_\mu D^\mu \Psi\]

where \[W^{1,2,3},B_\mu\] are the four spin 1 boson fields associated to the generators of the gauge transformation \[\Psi\]

The 3 generators of the \[SU(2)_L\] transformation are the three isospin operator components \[t^a=\frac{1}{2} \tau^a \] with \[\tau^a \] being the Pauli matrix and the generator of \[U(1)_\gamma\] being the weak hypercharge operator. The weak isospin "I" and hyper charge \[\gamma\] are related to the electric charge Q and given as

\[Q+I^3+\frac{\gamma}{2}\]

with quarks and lepton fields organized in left-handed doublets and right-handed singlets: 

For neutrinos involving Majorana mass an overview of the related mathematics is below including links to relevant papers

\[m\overline{\Psi}\Psi=(m\overline{\Psi_l}\Psi_r+\overline{\Psi_r}\Psi)\]

\[\mathcal{L}=(D_\mu\Phi^\dagger)(D_\mu\Phi)-V(\Phi^\dagger\Phi)\]

4 effective degrees of freedom doublet complex scalar field.

with 

\[D_\mu\Phi=(\partial_\mu+igW_\mu-\frac{i}{2}\acute{g}B_\mu)\Phi\]\

\[V(\Phi^\dagger\Phi)=-\mu^2\Phi^\dagger\Phi+\frac{1}{2}\lambda(\Phi^\dagger\Phi)^2,\mu^2>0\]

in Unitary gauge

\[\mathcal{L}=\frac{\lambda}{4}v^4\]

\[+\frac{1}{2}\partial_\mu H \partial^\mu H-\lambda v^2H^2+\frac{\lambda}{\sqrt{2}}vH^3+\frac{\lambda}{8}H^4\]

\[+\frac{1}{4}(v+(\frac{1}{2}H)^2(W_mu^1W_\mu^2W_\mu^3B_\mu)\begin{pmatrix}g^2&0&0&0\\0&g^2&0&0\\0&0&g^2&g\acute{g}\\0&0&\acute{g}g&\acute{g}^2 \end{pmatrix}\begin{pmatrix}W^{1\mu}\\W^{2\mu}\\W^{3\mu}\\B^\mu\end{pmatrix}\]

Right hand neutrino singlet needs charge conjugate for Majorana mass term (singlet requirement)

\[\Psi^c=C\overline{\Psi}^T\]

charge conjugate spinor

\[C=i\gamma^2\gamma^0\] 

Chirality

\[P_L\Psi_R^C=\Psi_R\]

mass term requires

\[\overline\Psi^C\Psi\] grants gauge invariance for singlets only.

\[\mathcal{L}_{v.mass}=hv_{ij}\overline{I}_{Li}V_{Rj}\Phi+\frac{1}{2}M_{ij}\overline{V_{ri}}V_{rj}+h.c\]

Higgs expectation value turns the Higgs coupling matrix into the Dirac mass matrix. Majorana mass matrix eugenvalues can be much higher than the Dirac mass.

diagonal of

\[\Psi^L,\Psi_R\] leads to three light modes v_i with mass matrix

\[m_v=-MD^{-1}M_D^T\]

MajorN mass in typical GUT 

\[M\propto10^{15},,GeV\]

further details on Majorana mass matrix

https://arxiv.org/pdf/1307.0988.pdf

https://arxiv.org/pdf/hep-ph/9702253.pdf

Now in order to account for the mass terms of DM the mass terms must be in or above the Kev range. Below are some related articles involving DESI. The Kev range would readily fall under the mentioned warm dm models. However there is also papers that place right hand neutrinos being in the GeV range through double beta decay.

DESI constraints

https://www.osti.gov/servlets/purl/3011043

Has a particular section to follow up on massive neutrinos behaving as dark matter described in above link.

https://arxiv.org/abs/2507.01380

double beta decay primer

https://arxiv.org/abs/2108.09364

In a nutshell the possibility is there so I started this thread to explore various examinations and starting a discussion on the the pros and cons of such a proposal. Naturally I would be interested in any related papers including counter arguments. This is not my own model proposal but a discussion on models presented by others. It doesn't suit a mainstream forum not yet anyways lol.

As for myself I see the potential but I question whether or not the mass terms will meet the required DM mass distribution. There was a fairly recent study that placed constraints on any simple Dirac mass term for right hand neutrinos in that examinations of the energy sector did not have any relevant findings. Still digging up that study hopefully I can find it however if I recall it constrained 5 KeV or less if memory serves.

other related papers

https://arxiv.org/pdf/1911.05092.pdf

https://arxiv.org/pdf/1901.00151.pdf

https://arxiv.org/pdf/2109.00767v2.pdf

https://arxiv.org/abs/1402.2301

https://arxiv.org/pdf/0708.1033

Located the light neutrino constraint paper via MicroBoone

https://arxiv.org/abs/2512.07159

Have you heard of the 2024 re-visit of Witten’s 1980s idea that DM could be composed of strangelets:

https://arxiv.org/html/2404.12094v1

I think this is very interesting, and perhaps warrants further investigation, since it requires no new particles to be hypothesized. There’s also this quite recent paper:

https://journals.aps.org/prd/abstract/10.1103/w1sd-v69d

which basically finds that large-scale geodesics are modified substantially if gravity is quantized on small scales, irrespective of the details of said quantisation.

I must admit this is the first time Ive heard of this particular possibility. Thank you for bringing it up. Lol knowing me I will dig considerably deeper into related articles to get a better feel for the status of Strangeness as DM +1

They would certainly drop out of thermal equilibrium early enough to form DM seeding for large scale structure formation

Maybe you guys can clear this to me?:
How particle origin of DM can explain Tully-Fisher Relation and Wide Binary's phenomena without shameless curve fitting? Isn't it require exactly the same evolution history of halos in vastly different galaxies? And where the halo is hiding in Wide Binary systems?

In so far as a particle origin for DM it would be easier to apply the virial theorem to DM with regards to galaxy rotation curves the weakly interactive characteristics for example well suits neutrinos. Or any other particle that is weakly interactive.

When you study the NFW profile which utilizes Virial theorem for galaxy rotation curves you find that its power law relations show that in order to obtain the measured galaxy rotation curves instead of the Keplar decline you need a greater amount of mass in fairly uniform distribution surrounding the galaxy. The DM halo distribution

while being weakly interactive DM is still subject to gravity. One of the primary reasons its expected to drop out of thermal equilibrium early is that its also considered responsible for initializing mass density anistropy for early large scale structure formation.

Obviously we cannot measure directly DM but we can certainly infer its existence in some form or another as a pressureless ( matter) component via the equations of state in cosmology.

Some of these articles describe secondary effects that can be measured involving DM. Example below.

The double beta decay in the above articles should also allow for some interactions that we can hopefully measure

https://arxiv.org/pdf/1402.4119

3 hours ago, Anton Rize said:

particle origin of DM

Personally, I’m not at all convinced that DM is necessarily particulate in nature. However, the possibility is still strong enough that it can’t be ruled out based on currently available data, so it’s good to explore what options there are in that regard. And not having to postulate any exotic new particles that are hard to fit into the Standard Model, is a big plus in my mind. Hence the reference to Witten’s idea.

Edited Thursday at 05:23 AM5 days by Markus Hanke

@Mordred thank you for detailed response

On 2/12/2026 at 1:23 PM, Mordred said:

One of the primary reasons its expected to drop out of thermal equilibrium early is that its also considered responsible for initializing mass density anistropy for early large scale structure formation.

So this seems to me like circular logic. Isn't it?:
1. We basically have 2 (rotation curves and structure formation) model-observation inconsistency's.
2. We patching them with phenomenology driven speculations
3. We trying to justify first speculation by using second speculation as a core of the argument.

I must have been missing something because this is clearly a methodological no no.

And also Im having major problems coherently structuring this physical process in my head:

On 2/12/2026 at 1:23 PM, Mordred said:

we can certainly infer its existence in some form or another as a pressureless ( matter) component via the equations of state in cosmology.

Its just
1. we have to assume the existence of unmeasured, very model dependent particle (not happy about it but ok...)
2. Then we have to assume some truly bizarre interaction protocols for this particle
3. Then we have to assume almost magical halo formation evolution that happened to have exact the same invariant relation with baryonic matter that would have to form exactly same way in Blue Compact Dwarfs (BDC) and in Intermediate Spiral (Sb)

And after all this assumptions and free parameters introduced we still cant explain all observations like Wide Binary's, some week lensing systems, structure formation, H_0, weak amplitude ($\ell=2$) in the CMB quadrupole moment, etc...

So my thinking is: "From my pov its all falling apart, but yet particle origin DM model remains dominant in modern cosmology. Therefor Im just missing something"

And problem is that the more I dig in to it I keep finding only new inconsistency's but not the explanation of why particle DM remains dominant theory.

It can't be that Im the first one who formulating this question like this right?

On 2/12/2026 at 3:23 PM, Markus Hanke said:

. However, the possibility is still strong enough that it can’t be ruled out based on currently available data,

Yes this is exactly what im trying to find. Can we pinpoint exactly why do we consider it a strong possibility?

Well it may help to consider that its not necessarily the galaxy rotation curves themselves that provide the strongest support of DM being a particle.

Consider the following if you take the FLRW metric and use the equations of state and apply the FLRW metric acceleration equations. Then remove the DM component and just apply baryonic matter of just 3% then there would never be enough matter in our universe for matter to become the dominant contributor to expansion.

Instead of radiation era, matter era the Lambda era. You would only go from radiation directly to Lambda dominant.

The Hubble constant would not have the value it does today. Matter radiation equality would never occur ( roughly when the universe is 7 Glyrs old.)

Expansion rates themselves and it how it evolves over time would be completely different.

Now as expansion occurs radiation diffuses more readily in an increased volume than matter so their densities evolve at different rates. Matter having an equation of state w=0 meaning it exerts no equivalent pressure term.

This one can construe as being the primary evidence that influences the research more in favor of a particle constituent.

Coupled with the detail that DM halos do cause gravitational lensing helps us confirm the density distributions. In point of detail Hubble telescope often makes use of these DM halos lenses to extend its range.

Hope that helps if you like some of the related mathematics I can post them here. Lol wouldn't take any real effort as I have em handy in another thread.

Edit correction on above the time frame was for matter lambda equality radiation/equality is sometime prior to Z=1150 depending on dataset used I would have to check later on.

Zeq 3387 using Planck 2018+BAO dataset.

@Mordred

That is a very specific point regarding the Equation of State ([math]w=0[/math]) and the necessity of a pressureless component to recover the correct expansion history and [math]H_0[/math].

You argued that without this particle component, the Universe would transition to [math]\Lambda[/math]-dominance incorrectly, making it impossible to match the observed Hubble constant and structure formation timing.

This is exactly the constraint I addressed in my open research. I found that if you strictly couple the vacuum geometry to the fine-structure constant ([math]\alpha[/math]) and Thermodynamics ([math]T_{CMB}[/math]), you derive [math]H_0[/math] analytically without any Dark Matter parameter:

[math]H_0 = \sqrt{8\pi G \frac{\rho_{\gamma}}{3\alpha^2}} \approx 68.15 \text{ km/s/Mpc}[/math]

This matches the Planck 2018 result ([math]67.4 \pm 0.5[/math]) within 1% purely from first principles.

Since your main argument for the "particle nature" is the necessity of matching these expansion constraints, does this exact geometric derivation of [math]H_0[/math] (which bypasses the need for a fitted [math]w=0[/math] component) count as counter-evidence in your view? Or do you see a physical flaw in linking the horizon scale to [math]\alpha[/math]?

Edited Saturday at 03:40 AM3 days by Anton Rize

15 hours ago, Anton Rize said:

Yes this is exactly what im trying to find. Can we pinpoint exactly why do we consider it a strong possibility?

Emphasis on “possibility”.

I think the reason why particulate DM is currently the favoured model is because it explains the widest range of observations in the simplest way with the least amount of extra assumptions. We can infer from observations that DM clumps around gravitational sources; it fits well to data concerning both the early universe, and current large-scale structure; it naturally explains observations around collisions of galaxy clusters; and at least some of these models fit well into the Standard Model.

Some of the other alternatives work better on specific subsets of the available data, but then fail on other subsets.

But again, I’m personally a bit sceptical, and my gut feeling is telling me that we’re missing something important here. Thus I’m looking forward to more research in this area.

It's weird that it's hidden how Galaxy shapes are just a manifestation of the very nature of fundamental particle composition...of course(fundamental particle) they are considered to be point like...I know it's controversial when I talk a bout fundamental particles composition.

And how Galaxy rotation curves are just linked to the very nature of dark matter...as i was learning about a simple harmonic oscillator,just a simple thing like Hooke's law,I was surprised to realize how Galaxy rotation is linked to dark matter more a kin to Hooke's law, with minor adjustments of constant k.

Excuse me for all this...I just find it hard to resist commenting...

2 hours ago, Markus Hanke said:

it explains the widest range of observations

1. Yes thats the explanation that I encounter most often. But when Im asking what observations exactly im getting bunch of model dependant non physical assumptions. So committed to the cause of scraping the bottom of this mystery well. Here's model independent raw observations that "Cosmological Dark Sector" boils down to according to my research:
   
   1. Orbital Speeds in Galaxies

2. Light Path Deflections Around Structures

3. Microwave Background Temperature Variations

4. Distant Supernova Flux Levels

5. Wide Binary Star Motions
   
   So as far as I understood accurate prediction and ontological explanation of all this phenomena, without introducing new entity's or speculations would solve the "Dark Sector Problem". I didn't include structure formation, its not directly observable, and here I just wanted to boil it down to the solid measurements. Do you guys agree with the list and criteria or am I missing something? @Markus Hanke @Mordred
   
   
   

   2 hours ago, MJ kihara said:

   Excuse me for all this...I just find it hard to resist commenting...

   We often blinded by the formalisms forgetting how mesmerizing Cosmos is. Thank you for reminding.

Your list above is fairly accurate though some of the list is fairly broad. Light path deflection for example would include spectography redshifting. For example integrated Sache Wolfe effect as signals pass through the mass variation of DM halos as one example.

If I think of anything not already covered on that list I will post it

As far as the fine structure constant your methodology from what you described here sounds remarkably similar to whats done in BSBM model (Berkenstein Model ) a version of TeVeS MOND.

The problem with coupling the fine structure constant is that you may find you would require a varying fine structure constant as per BSBM as well as the Hubble constant also varies over time. ( it's only constant everywhere at a given time slice. Ie today.

If you would like to test it at different Z ranges I can give you the Hubble constant value at any given redshift value. The cosmocalc in my signature which I was involved with developing has the correct second order terms for when the recessive velocity exceeds c for redshift beyond 1.49 ( Hubble Horizon) to the particle horizon.

the following below is for other readers to keep others at the same speed. The second order formula I'm referring to is the last formula on the list. The previous formulas is the mathematical proof using the equations of state and how they evolve over the universe expansion history.

FLRW Metric equations

\[d{s^2}=-{c^2}d{t^2}+a({t^2})[d{r^2}+{S,k}{(r)^2}d\Omega^2]\]

\[S\kappa(r)= \begin{cases} R sin(r/R &(k=+1)\\ r &(k=0)\\ R sin(r/R) &(k=-1) \end {cases}\]

\[\rho_{crit} = \frac{3c^2H^2}{8\pi G}\]

\[H^2=(\frac{\dot{a}}{a})^2=\frac{8 \pi G}{3}\rho+\frac{\Lambda}{3}-\frac{k}{a^2}\]

setting \[T^{\mu\nu}_\nu=0\] gives the energy stress mometum tensor as 

\[T^{\mu\nu}=pg^{\mu\nu}+(p=\rho)U^\mu U^\nu)\]

\[T^{\mu\nu}_\nu\sim\frac{d}{dt}(\rho a^3)+p(\frac{d}{dt}(a^3)=0\]

which describes the conservation of energy of a perfect fluid in commoving coordinates describes by the scale factor a with curvature term K=0.

the related GR solution the the above will be the Newton approximation.

\[G_{\mu\nu}=\eta_{\mu\nu}+H_{\mu\nu}=\eta_{\mu\nu}dx^{\mu}dx^{\nu}\]

Thermodynamics

Tds=DU+pDV Adiabatic and isentropic fluid (closed system)

equation of state

\[w=\frac{\rho}{p}\sim p=\omega\rho\]

\[\frac{d}{d}(\rho a^3)=-p\frac{d}{dt}(a^3)=-3H\omega(\rho a^3)\]

as radiation equation of state is

\[p_R=\rho_R/3\equiv \omega=1/3 \]

radiation density in thermal equilibrium is therefore

\[\rho_R=\frac{\pi^2}{30}{g_{*S}=\sum_{i=bosons}gi(\frac{T_i}{T})^3+\frac{7}{8}\sum_{i=fermions}gi(\frac{T_i}{T})}^3 \]

\[S=\frac{2\pi^2}{45}g_{*s}(at)^3=constant\]

temperature scales inversely to the scale factor giving

\[T=T_O(1+z)\]

with the density evolution of radiation, matter and Lambda given as a function of z

\[H_z=H_o\sqrt{\Omega_m(1+z)^3+\Omega_{rad}(1+z)^4+\Omega_{\Lambda}}\]

its other purpose was more my work testing the accuracy of the inverse relation to blackbody temperature. I rarely trust literature on any verbatim basis so often like to see how a statement such as temperature being the inverse of the scale factor is determined as being accurate. Sides its good practice lol ( above i had done previously in my Nucleosynthesis thread. )

the last formula the cosmocalc employs though has from version 1 of the cosmocalc well over a decade ago .

specifically this formula will provide the Hubble constant value as a function of redshift

\[H_z=H_o\sqrt{\Omega_m(1+z)^3+\Omega_{rad}(1+z)^4+\Omega_{\Lambda}}\]

should note for others as well the GR statements are for the Newton approximation which the FLRW metric falls under

just a side note the FLRW metric is not maximally symmetric where the Minkowskii metric under SR is. The use of the scale factor is one of the key issues with maximal symmetry (You can see this via the Christoffels for the FLRW metric ) or another way to learn this is through the Rayleigh equations.

Edited Sunday at 04:39 PM2 days by Mordred

On 2/16/2026 at 2:10 AM, Mordred said:

As far as the fine structure constant your methodology from what you described here sounds remarkably similar to whats done in BSBM model (Berkenstein Model ) a version of TeVeS MOND.

Yes I see how it could seem that way, but no. It has nothing to do with MOND. MOND is an embodiment of the exact phenomenological approach that I critic.

On 2/16/2026 at 2:10 AM, Mordred said:

The problem with coupling the fine structure constant is that you may find you would require a varying fine structure constant as per BSBM as well as the Hubble constant also varies over time. ( it's only constant everywhere at a given time slice. Ie today.

Good call that's exactly what I thought when I first derived it. But later it become clear that alpha is a geometrical invariant. Almost like pi. You can find the details here:

https://willrg.com/documents/WILL_RG_II.pdf#sec:invariant_alpha

In short it independently predicts H_0 and the CMB first peak \ell_1. So [math]\frac{\partial H_0}{\partial \ell_{peak}} = 0[/math]. And dynamically temperature changing [math]H(t) \propto T(t)^2 / \alpha[/math] but [math] \alpha=[/math]constant.

On 2/16/2026 at 2:10 AM, Mordred said:

If you would like to test it at different Z ranges I can give you the Hubble constant value at any given redshift value. The cosmocalc in my signature which I was involved with developing has the correct second order terms for when the recessive velocity exceeds c for redshift beyond 1.49 ( Hubble Horizon) to the particle horizon.

This is actually a very useful tool. Im constantly comparing my results with LCDM and this calculator will help me learn less python witch is great! thank you.

Edited 12 hours ago12 hr by Anton Rize

The formulas the calculator uses is on www.physicsforum.com under their insight article section if you go through the links you can get which formulas are employed for each column.

I still use it myself regularly as a cross reference and we continually test its accuracy for the particular datasets selected though it does allow a bit of additional adjustments outside of any particular dataset.

I should not in your article posted here the Saha equations give a range for example at 6000 kelvin you have 25% the neutral hydrogen at 3000 kelvin its roughly 75% and at 4000 kelvin its roughly 50%.

Deuterium is roughly 4500 kelvin if I recall for 75%. I would have to check later on. This pertains to your constraint mentioned in the article at 4000 kelvin.

I also question your statement of absolute coordinate time. Please explain as coordinate time is relative to the observer it isnt proper time.

Edit my memory was way off deuterium corresponds to 7.2 ×10 ^8 Kelvin for deuterium production.