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ASTROPHYSICAL BLACK HOLES AS A POTENTIAL SOURCE OF A SUBDOMINANT DARK-SECTOR COMPONENT

Energy screening, historical population-integrated throughput, and a phenomenological source-term framework

Preprint

Abstract

We examine whether energy released outside the horizons of accreting astrophysical black holes could produce a stable, subdominant dark-sector component. Production from the exterior plasma and extraction of rotational energy require explicit couplings. Energy that has crossed a classical event horizon cannot simply reappear outside because its products are invisible; any proposed horizon-mediated alternative needs a separate causal mechanism.

The paper deliberately does not claim that ordinary astrophysical black holes are the nature of dark matter. Instead, it formulates a narrower hypothesis: the existing cosmic BH population may provide an additional source term for dark matter or another stable dark-sector component.

The main object of analysis is defined by the equation

\[\dot\rho_X+3H(\rho_X+P_X)=Q_X,\qquad w_X=P_X/\rho_X\]

where \(X\) is a stable dark-sector component and \(Q_X\) is the cosmological source term. For a non-relativistic component

\[\dot{\rho}_{X}+3H\rho_X=Q_X\]

The source is expressed by integrating over the black-hole population:

\[Q_X(z)= \int dM\,d\chi\,n_{\rm BH}(M,\chi,z)\,\Gamma_{\rm BH}(M,\chi,z)\,Y_X(M,\chi,z)\,M c^2\]

where \(n_{\rm BH}\) is the physical mass-spin distribution, \(\Gamma_{\rm BH}\) is the effective fractional energy-conversion rate, and \(Y_X\) is the fraction of converted energy entering the component under consideration.

The framework distinguishes conversion efficiency, halo retention and particle survival:

\[\epsilon_X\equiv Q_X/(\dot\rho_{\rm BH,feed}c^2)\]
\[f_{\rm bound}\]

and

\[f_{\rm surv}\]

describing, respectively, conversion efficiency, the fraction of energy/particles remaining bound to the halo, and the fraction surviving to the epoch under consideration.

Energy availability is a quantitative constraint. A Soltan-type benchmark in section 28 gives a present fraction of about 1.4 × 10⁻⁶ times the conversion efficiency for the specified SMBH accretion budget. Kinematics, cosmological evolution, radiation and population constraints must then be checked separately.

We introduce the concept ofhistorical throughput:

\[\mathcal T_X(t)= \int_{t_{\rm form}}^{t} dt'\,Q_X(t') \left[\frac{a(t')}{a(t)}\right]^3 \mathcal S_X(t',t)\]

For cold particles, \(\mathcal S_X\) is the survival probability; it equals one when particles neither decay nor undergo any other removal process. Halo retention is not a loss from the cosmic inventory. Relativistic redshifting requires the momentum treatment of section 24.

This distinguishes a hypothesis dependent on the current BH density from one dominated by the integrated history of BH growth.

The paper does not introduce a specific microscopic theory of the dark sector. Instead, it constructs a minimal phenomenological framework in which the microphysics is compressed into a set of parameters

\[\{\Gamma_0,\alpha,\beta, \epsilon_X, f_{\rm bound}, f_{\rm surv}, m_X,\sigma_X,\ldots\}\]

For a subdominant component, we require

\[\Omega_X < \Omega_{\rm DM}\]

The previously considered exploratory interval

\[10^{-6}\lesssim \frac{\Omega_X}{\Omega_{\rm DM}} \lesssim 10^{-1}\]

is not an allowed region derived from data. Even its lower end is not reached with weak conversion in the SMBH benchmark above; smaller abundances must be included. The paper supplies an illustrative energy estimate, not a likelihood fit or evidence for a new component.

Thus, the proposed hypothesis can be turned from a qualitative idea into a testable model: the historical BH population must be reconstructed, \(Q_X(z)\) calculated, the produced component evolved, and CMB, large-scale structure, halo dynamics, indirect detection, cosmic backgrounds, and population constraints tested simultaneously.

1. Introduction

The modern cosmological model requires a cold or nearly cold invisible component that makes up most of the matter in the Universe. Planck cosmological data give

\[\Omega_c h^2\simeq 0.120\]

corresponding to the dominant non-baryonic matter component in the standard \(\Lambda\)CDM picture.

The origin of dark matter remains unknown.

Most standard approaches assume that dark matter was created in the early Universe and subsequently evolved mostly as nearly collisionless matter. However, another class of scenarios is logically possible:

Ordinary matter → an accreting BH system → hypothetical dark particles.

In that case, the dark sector need not be entirely a primordial relic population. Some fraction of it may continue to be produced after the formation of the first stars, galaxies, and black holes.

This possibility is the focus of the present work.

However, it is essential to distinguish two hypotheses from the outset.

Hypothesis A

Primordial black holes (PBHs) themselves are dark matter.

Hypothesis B

Ordinary astrophysical black holes are sources of an additional dark-sector component.

The present work primarily investigates hypothesis B.

This distinction is important.

PBHs may have formed in the early Universe and are indeed considered a possible dark-matter component; at the same time, their Hawking radiation can produce stable dark-sector particles. The literature shows that PBH evaporation can affect relic abundance and can even fully or partially produce dark matter in certain parameter ranges.

But astrophysical BHs have a different history:

\[M\sim 3M_\odot\;\ldots\;10^9M_\odot\]

stellar-mass BHs form through stellar evolution, while massive seeds can also arise through direct collapse. Subsequent growth involves accretion and mergers within evolving galactic structures. The mass range shown is illustrative, not an upper or lower census boundary.

Therefore, the dark-sector source in this case need not be Hawking evaporation.

The more natural phenomenological question is:

can a small fraction of the energy passing through a population of astrophysical black holes be systematically converted into stable dark-sector degrees of freedom?

This is the central hypothesis of this work.

2. Status of the hypothesis

The present model is aphenomenological source modelrather than a complete fundamental theory.

We do not claim the existence of a specific interaction

\[\mathcal L_{\rm int}\]

between the Standard Model and the dark sector.

Instead, we assume that some effective channel exists

The exterior environment of a BH → X, if a suitable interaction exists.

where \(X\) is a stable or sufficiently long-lived dark-sector particle.

In the most general form:

\[L_X=\epsilon_X\dot M_{\rm in}c^2\]

where

\[0\leq \epsilon_X\leq1\]

The key task is not to prove that \(\epsilon_X\neq0\), but to determine whether observations allow the range

\[0<\epsilon_X\ll1\]

This range is what makes the hypothesis physically interesting.

3. Why black holes are a natural target for this search

Active accreting or rotating BH systems can provide the following conditions; these properties do not apply to every BH:

- high energy density;

- strong gravitational redshift;

- high accretion rates during active phases;

- relativistic plasma;

- strong magnetic fields in accretion systems;

- relativistic jets;

- a large reservoir of rotational energy;

- horizon-scale physics.

In addition, the black-hole population has a substantial integrated growth history.

For SMBHs, the standard Soltan-type approach connects the integrated AGN luminosity density with the accumulated BH mass density. The observed local BH population is broadly consistent with a substantial fraction of SMBH mass having been accumulated through accretion.

Therefore, there is a natural quantity:

\[E_{\rm BH,history}=\int dt\,\dot E_{\rm acc}(t)\]

which has units of energy, whereas instantaneous luminosity has units of power. Their numerical values cannot be compared without specifying a timescale. The integral must also respect the measured growth and energy balance of the population.

This is why the analysis must focus not on an individual BH, but on the population-integrated source.

4. Energy screening

Throughout the paper, \(\rho_X\), \(\rho_{\rm DM}\) and \(P_X\) are physical energy densities and pressure, measured in the same units; \(Q_X\) is injected energy per physical volume per proper time. The BH density \(\rho_{\rm BH}\) is a mass density. A bar denotes a quantity per comoving volume, with \(a(t_0)=1\). The dimensionless BH spin is \(\chi=cJ/(GM^2)\); \(a(t)\) is reserved for the cosmological scale factor.

For an accreting object, define a total efficiency \(\epsilon_{\rm acc}\) for energy released outside the horizon. In the minimal benchmark this energy is partitioned between visible radiation and the dark sector, without additional mechanical losses or spin extraction:

\[L_{\rm available}=\epsilon_{\rm acc}\dot M_{\rm in}c^2,\qquad L_X=f_{\rm conv}L_{\rm available},\qquad L_{\rm vis}=(1-f_{\rm conv})L_{\rm available}\]
\[\dot M_{\rm BH}=(1-\epsilon_{\rm acc})\dot M_{\rm in},\qquad 0\le f_{\rm conv}\le1\]

Here the visible and dark luminosities share the same budget. An independently added dark loss, a jet or rotational extraction requires its own term in the balance. The rest energy \(\rho_{\rm BH}c^2\) is a scale, not an automatically extractable reservoir. Efficiencies must ultimately follow from a physical mechanism.

5. Three energy fractions

To avoid the most dangerous conceptual ambiguity, we introduce three distinct coefficients.

5.1. Conversion efficiency

\[f_{\rm conv}=\frac{E_{\rm converted}}{E_{\rm available}}\]

This is the efficiency of generating dark-sector energy.

5.2. Bound fraction

Not all produced energy must become part of a galactic halo.

Define

\[f_{\rm bound}(t)=N_{X,\rm bound}(t)/N_{X,\rm surviving}(t)\]

For relativistic production:

\[f_{\rm bound}\ll1\]

may be natural.

For a sufficiently cold component:

\[f_{\rm bound}\rightarrow1\]

may be possible.

Thus, energy generation and halo formation are different problems.

The numerator and denominator in this definition count bound and surviving particles of the same production cohort at the same epoch. Escape from a halo is not particle destruction.

5.3. Survival fraction

If \(X\) is unstable,

\[X\rightarrow Y+\cdots\]

then its lifetime must be taken into account:

\[\tau_X\]

We introduce

\[f_{\rm surv}(t',t)\]

Then the source contribution is:

\[d\rho_X(t;t')=Q_X(t')dt' \left(\frac{a(t')}{a(t)}\right)^3 f_{\rm surv}(t',t)\]

For a stable particle

\[f_{\rm surv}=1\]

6. Main source-term equation

For a cold dark-sector component:

\[\dot\rho_X+3H\rho_X=Q_X\]

with

\[Q_X(z)= \int dM\,d\chi\,n_{\rm BH}(M,\chi,z) \Gamma_{\rm BH}(M,\chi,z)\,Y_X(M,\chi,z) M c^2\]

where:

- \(M\) is the BH mass;

- \(\chi\equiv cJ/(GM^2)\) - dimensionless spin;

- \(n_{\rm BH}\) - BH mass-spin function;

- \(\Gamma_{\rm BH}\) - fractional source rate;

- \(Y_X\) - dark-sector energy yield.

The last two parameters can be combined:

\[\eta_X(M,\chi,z)=\Gamma_{\rm BH}Y_X\]

Then

\[Q_X= c^2\int dM\,d\chi\,n_{\rm BH} M\eta_X\]

Here \(n_{\rm BH}=dN/(dV_{\rm phys}\,dM\,d\chi)\); \(\Gamma_{\rm BH}\) and \(\eta_X=\Gamma_X=\Gamma_{\rm BH}Y_X\) have inverse-time units. \(Y_X\) is dimensionless. Rates can include an activity duty cycle. Empirical comoving number densities must be multiplied by \(a^{-3}\) before insertion in a physical \(Q_X\).

7. Population-integrated throughput

For an initially absent, stable, cold component, the homogeneous solution is

\[\rho_X(t)=\mathcal T_X(t)=a^{-3}(t)\int_{t_i}^{t}a^3(t')Q_X(t')\,dt'.\]

An initial abundance adds \(\rho_X(t_i)[a(t_i)/a(t)]^3\). For nonrelativistic particles with a constant rest-frame lifetime, the integrand is multiplied by \(\exp[-(t-t')/\tau_X]\). Relativistic production requires a momentum-dependent evolution; an arbitrary survival factor cannot replace it.

The halo-bound fraction does not multiply the mean cosmic density. Particles that leave a galaxy remain in the Universe. Their spatial distribution and contribution to a halo are separate transport problems. The present density depends on the source history, so it need not track current BH activity. This integral is the standard solution of a sourced continuity equation, not a new conservation law.

8. Historical memory

The same BH can undergo seed formation, accretion and mergers, so these are not disjoint populations that can simply be added. Instead, separate the physical source channels and avoid counting the same released energy twice:

Then:

\[Q_X= Q_{\rm seed} + Q_{\rm acc} + Q_{\rm merger}\]

For SMBHs, the accretion term is particularly natural:

\[Q_{\rm acc}=c^2\int dM\,d\chi\,n_{\rm BH}\dot M_{\rm in}\epsilon_{\rm acc}f_{\rm conv}\]

A merger-related contribution is also possible:

\[Q_{\rm merger} \propto \int dM_1\,dM_2\, R_{\rm merge} E_{\rm available} f_{\rm conv}\]

However, the merger channel is not taken as the primary one in this work.

The main benchmark is:

\[Q_X\propto \int n_{\rm BH}\dot M_{\rm acc}\,dM\,d\chi\]

9. Minimal benchmark model

To obtain testable scaling relations, introduce:

\[\Gamma_{\rm BH}=\Gamma_0 \left(\frac{M}{M_0}\right)^\alpha g(\chi) h(z)\]

and

\[Y_X=Y_0 \left(\frac{M}{M_0}\right)^\beta s(\chi) r(z)\]

Then:

\[Q_X=\Gamma_0Y_0c^2\int dM\,d\chi\,n_{\rm BH}M \left(\frac{M}{M_0}\right)^{\alpha+\beta} g(\chi)s(\chi)h(z)r(z)\]

If

\[\alpha+\beta=0\]

then the source approximately traces the BH mass density:

\[Q_X\propto\rho_{\rm BH}\]

If

\[\alpha+\beta>0\]

the weighting shifts toward larger masses at fixed population and spin weights. Actual dominance also depends on the mass function, its cutoffs and the activity distribution.

If

\[\alpha+\beta<0\]

then the contribution shifts toward smaller BHs.

This turns the model into a population-synthesis problem.

Only the sum \(\alpha+\beta\) is identified by this source weighting unless independent information separates the rate and yield. Spin and redshift factors must also be averaged consistently; \(\alpha+\beta=0\) does not by itself remove those dependences.

10. Black-hole mass function

Denote:

\[\Phi(M,z)=\frac{dN}{dV_{\rm phys}\,d\ln M}=M\int d\chi\,n_{\rm BH}(M,\chi,z)\]

Then:

\[\rho_{\rm BH}(z)=\int d\ln M\,M\Phi(M,z)\]

Source:

\[Q_X(z)=c^2\int d\ln M\,M\Phi(M,z)\Gamma_X(M,z)\]

This is a more convenient form for numerical analysis.

The SMBH mass function is one of the main empirical tools for reconstructing black-hole growth history.

11. Backreaction

A dark channel must draw energy from a specified reservoir. For a single object, an additional dark luminosity at fixed feeding and fixed other losses gives

\[\dot M_{\rm BH}=\dot M_{\rm standard}-L_{X,\rm extra}/c^2.\]

If dark emission instead replaces part of the existing radiative output, use the partition in section 4; subtracting it again would double-count the loss. Mergers and spin extraction require their corresponding mass and angular-momentum balances.

For the population, \(Q_X\) has units of energy per volume per time and cannot be added directly to a single-object mass rate. At the homogeneous level, donor components have compensating losses:

\[\dot\rho_i+3H(\rho_i+P_i)=Q_i,\qquad \sum_iQ_i=0.\]

All \(\rho_i\) in this equation are energy densities. A small perturbation requires the dark power to be small relative to the relevant feeding or extraction power, not relative to a net BH growth rate that can vanish.

12. Main dimensionless ratio

The most informative parameter is:

\[\epsilon_X(z)=\frac{Q_X(z)}{\dot\rho_{\rm BH,feed}(z)c^2}\]

If

\[\epsilon_X\ll1\]

this measures a small fraction of the supplied rest-mass energy. A small change to the radiative budget additionally requires \(\epsilon_X/\epsilon_{\rm acc}=f_{\rm conv}\ll1\) in the partition model; smallness must be tested against the relevant reservoir. It does not follow from the first inequality alone.

At the same time, the integrated contribution can be nonzero:

\[\frac{\rho_X}{\rho_{\rm DM}}=\frac{\int dt\,Q_Xa^3}{\rho_{\rm DM,0}a_0^3}\]

This allows the regime:

\[\epsilon_X\ll1, \qquad \Omega_X/\Omega_{\rm DM}\neq0\]

13. Why a subdominant component is the most natural scenario

Explaining all of dark matter through an astrophysical BH-source channel is considerably more difficult.

For a full component, one requires:

\[\Omega_X=\Omega_{\rm DM}\]

The source efficiency must then be sufficiently high while not violating:

- AGN luminosity functions;

- cosmic backgrounds;

- CMB energy injection;

- halo dynamics;

- structure formation;

- BH mass functions;

- indirect detection constraints.

For a subdominant component, the requirements are weaker:

\[\Omega_X=\xi\Omega_{\rm DM}, \qquad 0<\xi<1\]

An exploratory interval considered for comparison is:

\[10^{-6} \lesssim\xi\lesssim10^{-1}\]

This interval is not an inferred or preferred region. The weak-conversion SMBH benchmark in section 28 lies below it. Subdominance alone does not guarantee an adequate budget, cold kinematics or observable effects.

14. Why PBHs must be considered separately

PBHs have a completely different temporal kernel.

For a nonrotating, uncharged Schwarzschild BH, the Hawking temperature is:

\[T_H= \frac{\hbar c^3} {8\pi G M k_B}\]

that is,

\[T_H\propto M^{-1}\]

Therefore, small PBHs can efficiently produce heavy particles if

\[k_BT_H\gtrsim m_Xc^2\]

PBH evaporation is already used as a mechanism for producing dark matter and other beyond-SM states.

But for ordinary stellar and supermassive BHs:

\[k_BT_H\ll 1\,{\rm eV}\]

and Hawking emission is practically not an efficient mechanism for producing massive dark matter.

Therefore, the astrophysical-BH hypothesis must use a different source:

Accretion, rotation or the exterior environment → X.

rather than standard Hawking evaporation.

15. Possible microscopic channels

The phenomenological model does not fix the microphysics, but allows several classes of mechanisms.

15.1. Hidden-sector emission

Accretion plasma → X and other products,

if a weak coupling is present.

15.2. Gravitational production

A time-dependent gravitational background could in principle produce particles in a specified quantum-field model; a schematic stress tensor alone supplies neither a rate nor a production spectrum.

15.3. Horizon-associated channel

The process can be parameterized phenomenologically as:

A proposed horizon-associated process → X.

Such a mechanism requires a full quantum-gravity or semiclassical derivation and is therefore not treated as established in this work.

A weak electromagnetic coupling does not allow particles to cross a classical event horizon outward. An exterior plasma channel, superradiance and a proposed quantum-horizon process are separate mechanisms and must not share an assumed escape efficiency without a derivation.

15.4. Hidden-sector superradiance / rotational extraction

If the dark sector contains the relevant fields, BH rotational energy may be coupled to superradiant processes.

Then the source may depend on spin:

\[Q_X\propto g(\chi)\]

This makes the spin distribution an observable part of the model.

16. Spatial source term

A cosmological \(Q_X(z)\) is not sufficient to test the hypothesis at the galactic level.

We need to introduce:

\[Q_X(\mathbf x,t)=\int dM\,d\chi\,n_{\rm BH}(M,\chi,\mathbf x,t)\Gamma_XMc^2\]

Then:

\[\dot\rho_X+3H\rho_X+\frac{1}{a}\nabla\cdot(\rho_X\mathbf v_X)=Q_X\]

This equation allows the spatial correlation to be studied between:

- BH density;

- stellar mass;

- AGN activity;

- galaxy morphology;

- dark-matter halo.

17. Key distinction between source and tracer

Importantly:

\[\rho_X\not\propto n_{\rm BH}\]

in the general case.

The reason is transport.

If particles escape the host halo, not merely the immediate BH environment,

\[f_{\rm bound}\ll1\]

then the local source need not form a local halo.

Therefore, a kernel is required:

\[K(\mathbf x,t|\mathbf x',t')\]

Then:

\[\rho_X(\mathbf x,t)=\int_{t_i}^{t}dt'\,d^3x'\,K(\mathbf x,t|\mathbf x',t')Q_X(\mathbf x',t')\]

This turns the hypothesis into a transport + population-synthesis problem.

Use comoving positions \(\mathbf x\). The kernel includes expansion, the volume Jacobian and survival. For a stable cold homogeneous source it satisfies \(\int d^3x'K=[a(t')/a(t)]^3\), reproducing section 7. A halo-specific kernel can include capture or escape, but its integral is not the global cosmic inventory.

18. Perturbations

Let \(\Delta Q_X\) be the perturbation of the physical energy source in a specified gauge. For a fixed mass-spin bin, define fractional perturbations of the population, conversion rate and yield. To first order:

\[\Delta Q_X=c^2\int dM\,d\chi\,\bar n_{\rm BH}M\bar\Gamma_{\rm BH}\bar Y_X(\delta_n+\delta_\Gamma+\delta_Y).\]

A common effective contrast requires source-weighted averaging over the bins. The evolution of the density contrast also depends on metric perturbations, pressure, the injection momentum and compensating perturbations in the donor sector. A Green-function response must be derived from that closed system. A late source can modify the transfer function, but no transfer function or observational bound is computed here.

Only in this perturbation formula, bars on \(n,\Gamma,Y\) denote homogeneous background values, not conversion to comoving volume.

19. Effective bias

Define the source contrast \(\delta_q=\Delta Q_X/Q_X\) around the homogeneous nonzero source. A large-scale bias ansatz is

\[\delta_q\simeq b_Q(k,z)\delta_m.\]

The dark-component bias need not equal one and can depend on scale and time. For nonrelativistic matter, its contribution to the total matter contrast must use its fraction of all matter:

\[\delta_m=(1-F_X)\delta_{\rm rest}+F_X\delta_X,\qquad F_X=\rho_X/\rho_m.\]

Here \(\delta_{\rm rest}\) includes baryons and the remaining dark matter. The ratio \(\xi=\rho_X/\rho_{\rm DM}\) is not the weight in this total-matter equation. A change in the power spectrum requires solving the coupled perturbation equations; bias alone is not a prediction.

20. Early-Universe constraint

Ordinary stellar-remnant BHs form after stars and do not source a component during BBN or recombination. An earlier source requires a separate formation scenario and tests of expansion, donor depletion and perturbations during BBN, matter-radiation equality and recombination.

\[\rho_X(t_*)=a_*^{-3}\int_{t_i}^{t_*}a^3(t)Q_X(t)\,dt.\]

This is the recombination density for an initially absent stable cold component. It and the donor change must be compared with data. An integral without dilution, or comparison only with radiation, is insufficient. Late production still faces CMB lensing and structure constraints.

21. Late-time source

If

\[Q_X(z)\]

is maximal at

\[z\lesssim 10\]

then the produced component is intrinsically late-generated.

This distinguishes the model from a standard thermal relic.

However, late production should not automatically be interpreted as an advantage.

It is necessary to test:

\[\Delta N_{\rm eff}\]

CMB lensing,

matter power spectrum,

halo formation,

galaxy rotation curves,

satellite abundance.

22. Dark-sector temperature

If \(X\) is produced relativistically, its initial momentum distribution has the form:

\[\mathcal F_X(p,t_{\rm prod})\]

After redshifting:

\[p(t)=p_{\rm prod} \frac{a(t_{\rm prod})}{a(t)}\]

Free-streaming length:

\[\lambda_{\rm FS}=\int_{t_{\rm prod}}^{t_0} \frac{v(t)}{a(t)}dt\]

A warm or hot component can suppress small-scale structure; the size of the effect depends on both its abundance and momentum distribution.

The diagnostic quantity

\[\lambda_{\rm FS}\]

must therefore be tested jointly with the component fraction. A large free-streaming length alone does not exclude an arbitrarily small subcomponent.

23. Cold-production regime

To obtain dark-matter-like behavior, it is desirable to have:

\[\frac{p_{\rm prod}}{m_Xc} \ll1\]

or enough redshifting between production and the epoch being tested. A component produced after the first stars cannot become cold before matter-radiation equality, which occurred earlier.

If particles are produced with

\[p_{\rm prod}\gg m_Xc\]

they are initially radiation-like:

\[w_X\simeq\frac13\]

Then the equation is:

\[\dot\rho_X+4H\rho_X=Q_X\]

After the non-relativistic transition:

\[\dot\rho_X+3H\rho_X=Q_X\]

Therefore, a two-regime Boltzmann evolution must be used.

24. Minimal two-component evolution

A single phase-space distribution treats the relativistic-to-nonrelativistic transition continuously. Let \(\mathcal F_X(p,t)\) be the homogeneous isotropic occupation number, with physical momentum \(p\):

\[(\partial_t-Hp\partial_p)\mathcal F_X=\mathcal C_{\rm prod}-\mathcal C_{\rm loss}.\]
\[E=\sqrt{m_X^2c^4+p^2c^2},\qquad \rho_X=\int d\Pi\,E\mathcal F_X,\qquad P_X=\int d\Pi\,\frac{p^2c^2}{3E}\mathcal F_X.\]

The measure is \(d\Pi=g_Xd^3p/(2\pi\hbar)^3\), where \(g_X\) counts internal states. The energy moments give

\[\dot\rho_X+3H(\rho_X+P_X)=Q_X-D_X,\quad Q_X=\int d\Pi\,E\mathcal C_{\rm prod},\quad D_X=\int d\Pi\,E\mathcal C_{\rm loss}.\]

For stable particles with no annihilation, absorption or other removal, the loss term vanishes. The pressure approaches \(\rho_X/3\) in the relativistic limit and zero in the cold limit. Splitting the distribution into two bins is possible only with transfer terms that conserve number and energy; changing the injection label does not transfer already existing particles. No numerical phase-space evolution is performed here.

Here “stable and lossless” excludes annihilation, absorption and other removal processes as well as decay. Stability against decay alone does not set the collision loss term to zero.

25. Population synthesis

The practical calculation should be constructed as:

\[\Phi(M,\chi,z) \rightarrow \dot M(M,\chi,z) \rightarrow Q_X(M,\chi,z) \rightarrow \rho_X(z) \rightarrow P(k,z),C_\ell,\rho_X(r)\]

That is, one must model sequentially:

1. BH mass function;

2. spin distribution;

3. accretion history;

4. merger history;

5. source efficiency;

6. dark-sector spectrum;

7. transport;

8. cosmological evolution.

26. Benchmark population model

One can start with a separable approximation:

\[\Phi(M,\chi,z)=\Phi_M(M,z)P(\chi|M,z)\]

Then:

\[Q_X(z)=c^2\int d\ln M\,M\Phi_M(M,z)\Gamma_X(M,z)\]

For

\[\Gamma_X=\Gamma_0\left(\frac{M}{10^8M_\odot}\right)^\alpha(1+z)^\gamma\]

we obtain:

\[Q_X(z)=\Gamma_0c^2(1+z)^\gamma\int d\ln M\, M\Phi_M \left(\frac{M}{10^8M_\odot}\right)^\alpha\]

This is already a fully numerically implementable model.

In these equations \(\Gamma_X\) is the effective rate including the yield and the conditional spin average. It must not be multiplied by the yield a second time. The ansatz still requires a normalized spin distribution, a mass function, finite mass limits and a source budget.

27. Accretion-linked benchmark

A more physically transparent option is:

\[Q_X=f_{\rm conv}\epsilon_{\rm acc}\dot\rho_{\rm acc}c^2\]

Here:

\[\dot\rho_{\rm acc}=\int dM\,d\chi\,n_{\rm BH}\dot M_{\rm acc}\]

Then:

\[Q_X=f_{\rm conv}\epsilon_{\rm acc}\dot\rho_{\rm acc}c^2\]

This is particularly convenient because \(\dot\rho_{\rm acc}\) is related to AGN luminosity density through the Soltan argument.

Here \(\dot M_{\rm acc}=\dot M_{\rm in}\) is the rest-mass feeding rate, not net BH growth; \(\dot\rho_{\rm acc}\) is that rate per physical volume, not the total time derivative of the physical BH density. The efficiency is total released power as defined in section 4.

28. Order-of-magnitude energy condition

For a stable component born cold, the present energy density supplied by a constant conversion fraction is \(\rho_{X,0}=f_{\rm conv}\bar U_{\rm available}\), with \(\bar U_{\rm available}=\int a^3\dot u_{\rm available}\,dt\). The factor \(a^3\) converts the physical power density to power per comoving volume. Hence:

\[\xi_0=\frac{\rho_{X,0}}{\rho_{{\rm DM},0}},\qquad f_{\rm conv,req}=\frac{\xi_0\rho_{{\rm DM},0}}{\bar U_{\rm available}}.\]

If the required fraction exceeds one, this specified reservoir cannot supply the target abundance. A value below one passes only the energy test.

As an illustrative SMBH-only benchmark, assume accretion-dominated growth, negligible seed mass and merger losses, constant total release efficiency \(\epsilon_{\rm acc}=0.1\), and stable particles born cold. Use the classic local mass-density estimate of Marconi et al. [8], rescaled to \(h=0.674\), and the Planck base-ΛCDM value \(\Omega_ch^2=0.120\) [1]:

\[\rho_{{\rm BH},0}\simeq4.26\times10^5\,M_\odot\,{\rm Mpc}^{-3},\qquad \rho_{{\rm DM},0}/c^2\simeq3.33\times10^{10}\,M_\odot\,{\rm Mpc}^{-3}.\]
\[\bar U_{\rm available}/c^2\simeq\frac{\epsilon_{\rm acc}}{1-\epsilon_{\rm acc}}\rho_{{\rm BH},0}\simeq4.74\times10^4\,M_\odot\,{\rm Mpc}^{-3}.\]
\[\xi_0\simeq1.42\times10^{-6}f_{\rm conv}.\]

A conversion fraction of 0.01 gives \(\xi_0\simeq1.4\times10^{-8}\). The algebraic fraction needed for \(\xi_0=10^{-6}\) is about 0.70, outside the weak-conversion regime; it would also require refitting visible AGN luminosities and BH growth. This is not an allowed high-efficiency solution.

The quoted asymmetric uncertainty in [8] alone gives approximately \((1.0-2.0)\times10^{-6}\) for the coefficient multiplying the conversion fraction. It does not include uncertainty in efficiency, seeds, obscured growth or the local scaling relations. Alternative mass-density estimates can differ substantially [5]. Relativistic production loses additional energy through redshifting. Other BH populations and rotational or merger channels require their own budgets. This estimate therefore constrains this benchmark and establishes neither a universal upper bound nor a detection.

29. What a small \(f_{\rm conv}\) means

If the calculation gives, for example,

\[f_{\rm conv}\sim10^{-6}\]

this only means that obtaining the corresponding subcomponent requires converting one-millionth of the available energy throughput.

Such a result:

\[f_{\rm conv}\ll1\]

is not automatically evidence of physical realizability.

It must be shown that specific microphysics actually permits such a value.

30. Null models

The zero-source case is the null model. The remaining cases are competing source models, not independent null hypotheses. Their amplitudes, initial conditions and priors must be specified before comparison.

Model 1

\[Q_X=0\]

Standard \(\Lambda\)CDM.

Model 2

\[Q_X\propto\rho_{\rm BH}(z)\]

Model 3

\[Q_X\propto\dot\rho_{\rm acc}(z)\]

Model 4

\[Q_X(t)=\mathcal E_*\delta(t-t_*)\]

The amplitude \(\mathcal E_*\) is energy per physical volume at the injection time. The Dirac delta has inverse-time units, so \(Q_X\) has the required power-density units.

Model 5

PBH Hawking-production model.

This will show whether the data actually prefer a BH-history-dependent source.

31. Main model prediction

If the source is linked to accretion history:

\[Q_X(z)\propto\dot\rho_{\rm acc}(z)\]

then:

\[\rho_X(z)=\int^t dt'\,\dot\rho_{\rm acc}(t')c^2\epsilon_{\rm acc}f_{\rm conv} \left(\frac{a'}a\right)^3\]

Therefore, the dark component should have amemory kernel, linked to the history of AGN/BH growth.

This is the main distinguishing feature of the model.

32. Prediction for spatial correlation

If the source is linked to BHs:

\[Q_X(\mathbf x) \propto n_{\rm BH}(\mathbf x)\]

or

\[Q_X(\mathbf x) \propto \dot\rho_{\rm acc}(\mathbf x)\]

then a correlation arises between \(X\) and the galaxy/AGN population.

One can study the cross-power:

\[P_{X,g}(k,z)\]

If \(\xi\) is sufficiently large:

\[P_{X,g}\neq0\]

may become observable.

A nonzero galaxy correlation is not unique to this mechanism: ordinary cold dark matter also correlates with galaxies. A useful test must distinguish the predicted scale, redshift and amplitude dependence from standard clustering and astrophysical selection effects.

33. Prediction for halo profiles

If produced \(X\) remains gravitationally bound:

\[f_{\rm bound}\neq0\]

then the source is strongest in regions of high BH density.

However, this does not automatically imply a central spike.

Transport can smooth the profile:

\[\rho_X(r,t)=\int_{t_i}^{t}dt'\int_0^\infty4\pi r'^2dr'\,K_r(r,t|r',t')Q_X(r',t')\]

Therefore, central enhancement must be calculated, not assumed.

For this local halo expression, \(r\) and \(r'\) are physical radii and the kernel includes any required volume transformation, transport and survival. It is not the same normalization convention as the comoving-coordinate kernel in section 17.

34. Possible connection to galactic morphology

If production is linked to AGN activity, then:

\[Q_X \sim Q_X({\rm AGN})\]

Then elliptical galaxies and massive bulges may have enhanced source histories.

However, current morphology dependence is too complex to turn into a specific prediction without population synthesis.

Therefore, in this work this remains a secondary observable.

35. Gravitational-wave channel

BH mergers produce:

\[E_{\rm GW} \sim \eta_{\rm GW}Mc^2\]

If dark-sector conversion is linked to merger dynamics:

\[E_X=f_{\rm merger}E_{\rm available}\]

then:

\[Q_X^{\rm merger}=\int dM_1\,dM_2\,R_{\rm merge}f_{\rm merger}E_{\rm available}\]

However, the current GW population does not require such an additional channel.

Therefore, the merger channel should be treated as a separate testable extension.

Analyses of LVK data also strongly constrain the possibility that the observed stellar-mass BH population is substantially PBH; for example, one O3 analysis obtains \(f_{\rm PBH}\lesssim10^{-3}\) in the range \(1-200M_\odot\) for the scenarios considered.

The merger rate in the source integral is per physical volume, source proper time and component-mass intervals at a fixed redshift. Converting an observed event-rate distribution requires volume and time-dilation factors; integrating over redshift belongs to the subsequent cosmic history calculation. PBH abundance limits depend on the assumed mass function, binary formation and clustering and are not a bound on this dark-emission efficiency.

36. Electromagnetic constraints

If dark-sector production is accompanied by visible-sector emission:

\[X\rightarrow\gamma+\cdots\]

then one must account for:

\[\Delta I_\gamma(E)\]

cosmic X-ray background,

gamma-ray background,

radio backgrounds.

Therefore:

\[f_{\rm conv}\]

cannot be considered independently of branching ratios.

37. CMB constraints

Visible energy injection can alter ionization, gas temperature and CMB anisotropies. Prompt emission accompanying production and delayed particle decay are different sources. For nonrelativistic particles of rest-frame lifetime \(\tau_X\), a visible decay branching fraction gives

\[Q_{\rm vis,dec}=f_{\rm vis}\rho_X/\tau_X.\]

Deposition into the gas is not generally instantaneous or complete. It requires an energy- and redshift-dependent transfer calculation. Constraints must come from the resulting ionization and CMB histories, rather than a universal pointwise ceiling on \(Q_X(z)\). Even a fully invisible component affects expansion and gravitational perturbations.

38. Large-scale structure

For a cold subcomponent:

\[\delta_X\]

may be close to the standard CDM perturbation.

For a warm component:

\[\lambda_{\rm FS}\]

may suppress small-scale power.

Therefore, the model must satisfy:

\[\left| \frac{\Delta P(k)} {P_{\Lambda{\rm CDM}}(k)} \right| < \varepsilon_P(k,z)\]

39. Halo abundance

If \(X\) appears late, it may not have time to participate in early halo formation.

For a stable cold component with a nonnegative source and a conserved cold reference component, the fraction grows toward the present:

\[\xi(z_{\rm form})\le\xi(z_0)\]

This leads to a potential difference between:

- total matter today;

- matter available during early structure formation.

This effect is one of the strongest ways to test a late-source scenario.

40. Local Milky Way constraint

For our Galaxy:

\[Q_X(\mathbf r)=\sum_iL_{X,i}\delta^3(\mathbf r-\mathbf r_i)\]

The main contribution may come from the central SMBH or from the entire stellar-mass BH population.

But one should not assume:

\[Q_X\sim Q_{\rm SMBH}\]

without a population calculation.

If stellar BHs are numerous, their integrated contribution may compete with that of the SMBH.

41. Stellar-mass versus supermassive BH

The model must distinguish:

\[Q_X=Q_X^{\rm stellar}+Q_X^{\rm intermediate}+Q_X^{\rm SMBH}\]

If:

\[\Gamma_X\propto M^\alpha\]

then the relative contribution is determined not only by object number but also by mass weighting:

\[Q_i\propto\int d\ln M\,M^{1+\alpha}\Phi_i(M)\]

This makes the exponent \(\alpha\) critical.

42. Spin dependence

If:

\[\Gamma_X=\Gamma_0g(\chi)\]

then:

\[Q_X \propto \langle g(\chi)\rangle\]

For a superradiant mechanism:

\[g(\chi)\]

may increase sharply with spin.

The abundance then depends on the spin history through the source integral. Its density parameter is defined by:

\[\Omega_X(z)=\rho_X(z)/(\rho_{\rm crit}(z)c^2)\]

Here \(\rho_{\rm crit}=3H^2/(8\pi G)\) is the critical mass density; the factor \(c^2\) converts it to energy density. This definition alone is not an observable prediction.

43. Information encoded in the BH population

Thus, the BH population acts as a kind of cosmic integrator:

\[\{M,\chi,\dot M,z\} \rightarrow Q_X\]

The current BH catalogue contains incomplete information about the past.

Therefore, reconstruction is required:

\[P(M,\chi,\dot M|z)\]

This ishistorical population synthesis.

44. Historical throughput versus current activity

Consider two scenarios.

Model A

\[Q_X(z)\propto\rho_{\rm BH}(z)\]

Model B

\[Q_X(z)\propto\dot\rho_{\rm acc}(z)\]

They can give the same \(Q_X(z=0)\), but completely different:

\[\rho_X(z)\]

Therefore, measuring only today’s BH density is insufficient.

It is necessary to reconstruct:

\[Q_X(z)\]

throughout cosmic history.

45. Relation to AGN luminosity functions

Let \(\bar\Phi_L(L,z)=dN/(dV_{\rm com}\,d\ln L)\) be the bolometric luminosity function of visible AGN, corrected for selection and obscuration. In the partition model of section 4, for constant \(0\le f_{\rm conv}<1\),

\[Q_X(z)=a^{-3}\frac{f_{\rm conv}}{1-f_{\rm conv}}\int d\ln L\,\bar\Phi_L(L,z)L.\]

The factor \(a^{-3}\) converts a comoving luminosity density to a physical source density. For weak conversion, the ratio is approximately \(f_{\rm conv}\). For a luminosity-dependent efficiency it belongs inside the integral. High conversion cannot be inferred while keeping both the visible luminosity function and the standard BH growth calibration unchanged.

46. Soltan-type consistency test

For a constant total release efficiency, the accretion contribution to the comoving BH mass growth obeys

\[\frac{d\bar\rho_{{\rm BH},\rm acc}}{dt}=(1-\epsilon_{\rm acc})\dot{\bar\rho}_{\rm in}.\]

The corresponding physical dark source is

\[Q_X=a^{-3}f_{\rm conv}\frac{\epsilon_{\rm acc}}{1-\epsilon_{\rm acc}}c^2\frac{d\bar\rho_{{\rm BH},\rm acc}}{dt}.\]

Here the derivative refers only to accreted mass accumulated per comoving volume; seed formation, merger losses and ejected objects require separate accounting. Visible radiation has efficiency \((1-f_{\rm conv})\epsilon_{\rm acc}\). Thus AGN luminosities and BH demographics jointly constrain the partition. Hidden or radiatively inefficient growth introduces systematic uncertainty, not an unlimited energy reservoir [5, 6, 7, 9].

47. Fundamental limitation

Even if:

\[f_{\rm conv}\ll1\]

is energetically sufficient, this still does not prove the existence of a dark-sector coupling.

Passing an energy-budget test does not establish a microscopic production mechanism.

This is a fundamental limitation of the present framework.

48. Falsification criteria

The hypothesis should be considered excluded in a given parameter region if at least one of the following holds:

1. Energetic impossibility

\[f_{\rm conv}>1\]

2. CMB violation

A predicted ionization or CMB history excluded by the joint data analysis.

3. Structure violation

\[\Delta P(k)\]

exceeds the allowed range.

4. Free-streaming violation

\[\lambda_{\rm FS}\]

is too large.

5. BH-demography violation

Backreaction substantially changes the observed BH mass function.

6. Background violation

Produced visible secondaries exceed cosmic backgrounds.

7. Population inconsistency

There is no allowed

\[\Phi(M,\chi,z)\]

that gives the required \(Q_X\).

These criteria exclude specified parameter choices after a joint data analysis. A free efficiency approaching zero cannot be excluded by nondetection alone. Free-streaming and visible-energy limits depend on abundance, spectra and deposition; there is no universal threshold independent of those assumptions.

49. Positive test

A positive result should not look like:

“we found dark matter in black holes.”

It should instead be formulated much more strictly:

\[D \Rightarrow Q_X(z) \neq0\]

and simultaneously:

\[Q_X(z)\]

must be statistically associated with

\[\Phi_{\rm BH}(M,\chi,z)\]

or

\[\Phi_{\rm AGN}(L,z)\]

That is, one must detect aBH-history-dependent dark component.

Such evidence would require a statistically preferred nonzero source in a specified model and rejection of plausible alternatives, including ordinary galaxy bias. A correlation by itself does not identify the microscopic origin.

50. Statistical framework

Parameters:

\[\theta= \{ f_{\rm conv},\alpha,\beta,\gamma, m_X,\tau_X, f_{\rm bound}, f_{\rm vis} \}\]

Likelihood:

\[\mathcal L(\theta)=\mathcal L_{\rm CMB}\mathcal L_{\rm LSS}\mathcal L_{\rm BH}\mathcal L_{\rm GW}\mathcal L_{\rm EM}\]

Posterior:

\[P(\theta|D) \propto \mathcal L(D|\theta) \pi(\theta)\]

Main parameter:

\[\xi= \frac{\Omega_X}{\Omega_{\rm DM}}\]

The factorized likelihood is valid only for conditionally independent datasets; overlapping samples and shared systematics require a joint covariance. \(\xi\) is derived from the source normalization and evolution, not an additional independent abundance to tune. Degenerate rate/yield parameters need independent information or a reduced parameter set.

51. Hierarchical population inference

Instead of a fixed BH mass function:

\[\Phi(M,z)\]

it should be treated as a latent population model:

\[\Phi(M,z|\psi)\]

where \(\psi\) denotes the parameters of population evolution.

Then:

\[P(\xi,\psi|D) \propto P(D|\xi,\psi) P(\xi)P(\psi)\]

This allows uncertainty in BH demography to be included.

The product of priors assumes prior independence. If abundance is derived from source and population parameters, use their joint prior and induced abundance distribution instead of assigning an unrelated prior to the same quantity twice.

52. Minimal numerical experiment

The minimal numerical pipeline should include:

\[\Phi(M,z) \rightarrow \dot M(M,z) \rightarrow Q_X(z) \rightarrow \rho_X(z) \rightarrow \xi(z)\]

followed by the calculation of:

\[P(k,z)\]
\[C_\ell^{TT,TE,EE,\phi\phi}\]

and halo observables.

53. Benchmark parameter scan

At minimum, one should investigate:

\[10^{-8} \le f_{\rm conv} \le 10^{-1}\]
\[-2\le\alpha\le2\]
\[-4\le\gamma\le4\]

and for the dark-sector particle:

\[10^{-6}\,{\rm eV} \le m_Xc^2 \le 10^6\,{\rm GeV}\]

This is not a claim that the entire range is allowed; it is the range for an initial numerical scan.

These are exploratory coordinates, not a demonstrated accessible mass range. A specific production mechanism determines thresholds and spectral support; a keV plasma cannot thermally populate arbitrary heavy masses. Include the zero-source case and scan smaller conversion fractions where relevant. No parameter scan has been performed in this paper.

54. Main expected regimes

Regime I - negligible

\[\xi\ll10^{-6}\]

Cosmologically small abundance; detectability still depends on the interaction and observable, not on this fraction alone.

Regime II - weak subcomponent

\[10^{-6}\lesssim\xi\lesssim10^{-3}\]

An exploratory interval, not a natural consequence of weak coupling. The weak-conversion SMBH benchmark falls below it.

Regime III - appreciable subcomponent

\[10^{-3}\lesssim\xi\lesssim10^{-1}\]

Requires a separately demonstrated energy reservoir as well as stronger observational tests. It is not supported by the benchmark budget in section 28.

Regime IV - dominant

\[\xi\sim1\]

The most stringent scenario and not the main target of this work.

55. Why \(\xi\sim1\) is not the central goal

For a dominant component, one must reproduce:

\[\Omega_{\rm DM}\]
\[P(k)\]
\[C_\ell\]

halo abundance,

galaxy dynamics

simultaneously.

The timing of production is a separate constraint.

If most DM appears too late, it does not participate in early structure formation.

Therefore, the subdominant scenario is a substantially more conservative hypothesis.

56. Comparison with PBH dark matter

In the first case, the source history is determined by early cosmology.

In the second:

\[Q_X(z)\]

follows the history of star formation, galaxy formation, AGN activity, and BH growth.

These are fundamentally different observational signatures.

Primordial production: the early Universe creates PBHs; surviving PBHs can themselves supply part of dark matter, while an evaporating population can produce new particles. Astrophysical production: stellar evolution or massive-seed formation creates BHs whose later environment may supply X.

57. Relation to hidden-sector physics

Hidden-sector dark matter is already studied in theories where dark particles interact with the visible sector very weakly or predominantly gravitationally.

A particularly interesting scenario is one in which ordinary astrophysical objects act as mediators between the sectors.

In this sense, the BH source model can be viewed as a form of:

an astrophysical connection between the visible and hidden sectors.

But the specific portal must be specified by a separate particle-physics model.

58. Relation to existing work and scope of novelty

Dark-particle production near accreting astrophysical BHs has already been studied, including thermal plasma processes, acceleration of existing dark particles and superradiance. Cai et al. [14] is directly relevant, in addition to the PBH literature. Acceleration of existing particles is not creation of a new rest-mass inventory.

The present framework combines population history, conversion, survival and transport as an accounting scheme. Time integration of a source is standard. No priority claim follows from writing this integral. A distinct contribution would require a calculated population yield, a new bound or an observable signature compared quantitatively with existing models.

59. Strongest version of the hypothesis

The strongest formulation of the hypothesis is:

The conditional hypothesis is that a component \(X\), if produced by a specified astrophysical-BH mechanism, has an abundance determined by the source history. For initially absent, stable cold particles this takes the form below, with \(\mathcal S_X=1\). A finite nonrelativistic lifetime adds the survival factor of section 7.

Mathematically:

\[\rho_X(t)=\int_{t_{\rm form}}^t dt'\int dM\,d\chi\,n_{\rm BH}(M,\chi,t')\,\Gamma_{\rm BH}(M,\chi,t')\,Y_X(M,\chi,t')Mc^2\left[\frac{a(t')}{a(t)}\right]^3\mathcal S_X(t',t)\]

This is the central equation of the model.

60. Optimistic energy benchmark

For the minimal model, take:

\[Y_X=1\]
\[f_{\rm bound}=1\]
\[f_{\rm surv}=1\]

but interpret them asoptimistic upper bound.

Then:

\[Q_X^{\rm max}=f_{\rm conv}\dot u_{\rm available}\]

If even this upper bound cannot produce the required \(\xi\), the model is excluded.

If it does, the second stage begins:

\[Y_X<1\]

transport,

free-streaming,

decay,

visible branching.

61. Upper-bound logic

Thus, the analysis should proceed in the following order:

Energy budget → population constraints → cosmological evolution → transport, with a microscopic mechanism needed to connect these steps.

This prevents over-interpretation of the result.

62. Possible outcome of numerical analysis

Three situations may arise.

Case A

\[f_{\rm conv}>1\]

The hypothesis is energetically impossible.

Case B

\[10^{-10}<f_{\rm conv}<10^{-3}\]

The required fraction passes the energy test for the chosen abundance and reservoir. A microscopic mechanism, a spectrum and observational consistency still need to be established; no generic coupling strength follows from this number.

Case C

\[f_{\rm conv}\sim10^{-1}-1\]

The hypothesis becomes substantially more constrained and requires a very efficient dark-sector coupling.

63. Required continuation of the work

The next stage should be computational rather than conceptual.

It is necessary to build:

Stage 1

Empirical BH mass function.

Stage 2

Spin distribution.

Stage 3

AGN luminosity function.

Stage 4

Historical accretion density.

Stage 5

Population-integrated \(Q_X(z)\).

Stage 6

Boltzmann evolution.

Stage 7

Matter power spectrum.

Stage 8

CMB/LSS likelihood.

64. Expected form of the result

The result should be a constraint:

\[\xi < \xi_{\rm max}(m_X,\alpha,\gamma,f_{\rm vis},\ldots)\]

rather than a single arbitrary number.

For example:

\[\xi_{\rm max}=\xi_{\rm max}(m_X,\alpha,\gamma)\]

This makes the result suitable for publication and subsequent comparison with other dark-sector models.

65. Main observational discriminator

The most interesting test:

the relation between a hypothetical dark component and BH growth history.

If the produced component results from a primordial mechanism, its abundance is determined by the early Universe.

If:

\[Q_X\propto\dot\rho_{\rm acc}\]

then it should be linked to cosmic AGN history.

a quantitatively distinct correlation with BH/AGN history, after accounting for ordinary galaxy clustering.

66. Limitations of this preprint

The limitations should be stated explicitly.

1.

No specific fundamental interaction is derived.

2.

The existence of dark-sector emission from astrophysical BHs is not demonstrated.

3.

A complete numerical population synthesis has not been performed.

4.

A Boltzmann/CMB likelihood analysis has not been performed.

5.

The exact \(f_{\rm bound}\) has not been calculated.

6.

The production spectrum \(dN_X/dp\) has not been specified.

7.

Not all alternative sources have been excluded.

This remains a phenomenological research framework with an illustrative energy-budget calculation. No dark component is detected, no microscopic production rate is derived, and no cosmological parameter region is claimed to have passed a full likelihood analysis.

67. What the paper claims

The supported result is a conditional budget: for the stated SMBH accretion benchmark, \(\xi_0\simeq1.42\times10^{-6}f_{\rm conv}\). This makes a cosmologically appreciable abundance difficult at weak conversion. It does not establish that the conversion exists. Whether any specified interaction produces stable particles with the required spectrum, survival and clustering remains open.

68. What the paper does NOT claim

BHs are not identified here with all dark matter or with white holes. No daughter universe or quantum-gravity process is established by this framework.

69. Speculative extensions

Transitions from a black hole to a white hole, daughter universes and topology-changing processes are speculative extensions. They require a causal and dynamical model beyond this work and are not used in the energy estimate or the main source equations.

70. Conclusion

A phenomenological framework has been proposed to test the hypothesis that astrophysical black holes can be a source of a subdominant dark-sector component.

Key equation:

\[\dot\rho_X+3H\rho_X=Q_X\]

with a population-integrated source:

\[Q_X=c^2\int dM\,d\chi\,n_{\rm BH}M\Gamma_{\rm BH}Y_X\]

Historical component density:

\[\rho_X(t)=\int^t dt'\,Q_X(t')\left(\frac{a'}a\right)^3\mathcal S_X\]

This leads to the central conclusion:

The present BH population and its historical energy release are different quantities connected by a common growth and energy budget.

A test must use the population and its growth history together. Present BH mass density remains a valuable integral constraint through energy accounting; it cannot simply be discarded in favor of a historical reservoir.

\[\Phi_{\rm BH}(M,\chi,z)+\dot M_{\rm acc}(M,\chi,z)\rightarrow Q_X(z)\rightarrow\rho_X(z)\]

The illustrative SMBH budget yields \(\xi_0\simeq1.42\times10^{-6}f_{\rm conv}\), so a one-percent conversion gives about \(1.4\times10^{-8}\) of the reference dark-matter density. Broader source channels require additional calculations. This estimate neither demonstrates a microscopic channel nor establishes an observationally allowed model.

Therefore, the central scientific question is formulated as follows:

Can a specified, causally consistent interaction convert the available energy into stable particles that remain sufficiently cold and satisfy population, electromagnetic, gravitational and cosmological constraints?

This question is concrete, quantitative, and potentially falsifiable.

Appendix A. Equations and required closure

For a stable homogeneous component, use the energy equation and the population-integrated source:

\[\dot\rho_X+3H(\rho_X+P_X)=Q_X,\qquad Q_X=\int dM\,d\chi\,n_{\rm BH}L_X.\]

For the physical mass-spin number density, the schematic population continuity equation is

\[\partial_t n_{\rm BH}+3Hn_{\rm BH}+\partial_M(\dot M n_{\rm BH})+\partial_\chi(\dot\chi n_{\rm BH})=S_{\rm seed}+S_{\rm merger,gain}-S_{\rm merger,loss}.\]

Seed formation and merger gain/loss terms must avoid double counting; merger kernels, spin evolution and boundary conditions still have to be supplied. A comoving population \(\bar n=a^3n\) removes the explicit dilution term after all terms are transformed consistently.

In the minimal accretion partition:

\[\dot M_{\rm BH}=(1-\epsilon_{\rm acc})\dot M_{\rm in},\qquad L_X=f_{\rm conv}\epsilon_{\rm acc}\dot M_{\rm in}c^2.\]

Together with the phase-space equation in section 24 and compensating donor losses in section 11, these are equations to be closed by a specified mechanism and population model. They are not yet a solved cosmological model.

Appendix B. Dimensionless formulation

Define:

\[x=\ln a\]

Then:

\[\frac{d\rho_X}{dx}+3\rho_X=\frac{Q_X}{H}\]

For the comoving density:

\[\bar\rho_X=a^3\rho_X\]

we obtain:

\[\frac{d\bar\rho_X}{dt}=a^3Q_X\]

Therefore:

\[\bar\rho_X(t)=\int^t a^3(t')Q_X(t')dt'\]

This is the most convenient form for numerical integration.

The displayed integral assumes zero initial abundance, stable cold particles and no losses; otherwise add the initial term and the appropriate loss or momentum evolution. In redshift variables use \(dt=-dz/[(1+z)H(z)]\), with reversed integration limits.

Appendix C. Effective source efficiency

Define:

\[\epsilon_{\rm eff}(z)=\frac{Q_X}{\dot\rho_{\rm acc}c^2}\]

Then:

\[Q_X=\epsilon_{\rm eff}\dot\rho_{\rm acc}c^2\]

If \(\epsilon_{\rm eff}\) is constant:

\[\rho_X(t)=\epsilon_{\rm eff}\int^t dt'\,\dot\rho_{\rm acc}(t')c^2\left(\frac{a'}a\right)^3\]

If it depends on mass:

\[\epsilon_{\rm eff}=\epsilon_0\left(\frac{M}{M_0}\right)^\alpha(1+z)^\gamma g(\chi)\]

Appendix D. Benchmark-model parameter set

Minimal parameter set:

\[\theta=\{\epsilon_0,\alpha,\gamma,m_X,\tau_X,f_{\rm bound},f_{\rm vis}\}\]

Additionally:

\[\beta\]

for a mass-dependent yield and

\[\chi_0\]

for spin dependence.

Appendix E. Main viability criterion

For the proposed subdominant conversion scenario require \(0\le\xi<1\), a physically available source budget and a mechanism with allowed efficiencies. Compare the full predicted data vector, including power spectra, CMB, visible backgrounds and BH demographics, with a joint likelihood including covariance and nuisance parameters. Pointwise bounds on spectra or injected power are only approximations tied to a specified model. Passing these tests would establish compatibility, not discovery or uniqueness.

Appendix F. Working classification of scenarios

Scenario

Main source

Time dependence

Main test

PBH evaporation

Hawking radiation

early Universe

BBN/CMB/gamma

Astrophysical accretion

AGN/BH growth

\(z\sim0-10\)

LSS/CMB/BH history

BH merger

merger energy

episodic

GW correlation

Spin extraction

BH rotation

spin-dependent

BH spin population

Horizon microphysics

unknown

model-dependent

multi-messenger

Appendix G. Main falsifiable prediction

If the model is realized through an accretion-linked source:

\[Q_X(z)\propto\dot\rho_{\rm acc}(z)\]

then the abundance of \(X\) should have a definite temporal kernel:

\[K_X(z,z')\]

Therefore, different dark-sector observables should be consistent with the same function:

\[K_X(z,z')\]

This makes it possible to move from a simple claim about the possibility of BH-produced dark matter to quantitative hypothesis testing.

References and key guideposts

  1. Planck Collaboration. Planck 2018 results. VI. Cosmological parameters. arXiv:1807.06209.
  2. J. Auffinger. Primordial black hole constraints with Hawking radiation - a review. arXiv:2206.02672.
  3. N. Bernal, Ó. Zapata. Dark Matter in the Time of Primordial Black Holes. arXiv:2011.12306.
  4. P. Gondolo, P. Sandick, B. Shams Es Haghi. The Effects of Primordial Black Holes on Dark Matter Models. arXiv:2009.02424.
  5. A. Comastri et al.. Mass without radiation: heavily obscured AGN, the X-ray Background and the Black Hole Mass Density. arXiv:1501.03620.
  6. Q. Yu, Y. Lu. Toward precise constraints on growth of massive black holes. arXiv:0808.3777.
  7. A. Merloni. Cosmological evolution of supermassive black holes and AGN: a synthesis model for accretion and feedback. arXiv:0808.0070.
  8. A. Marconi et al.. Local Supermassive Black Holes, Relics of Active Galactic Nuclei and the X-ray Background. arXiv:astro-ph/0311619.
  9. K. Jahnke. The Soltan argument at z=6: UV-luminous quasars contribute less than 10% to early black hole mass growth. arXiv:2411.03184.
  10. M. Andrés-Carcasona et al.. Constraints on primordial black holes from LIGO-Virgo-KAGRA O3 events. arXiv:2405.05732.
  11. R. Foot, S. Vagnozzi. Dissipative hidden sector dark matter. arXiv:1409.7174.
  12. A. Boudon et al.. Baryogenesis through Asymmetric Hawking Radiation from Primordial Black Holes as Dark Matter. arXiv:2010.14426.
  13. K. Agashe et al.. Light in the Shadows: Primordial Black Holes Making Dark Matter Shine. arXiv:2409.13811.
  14. R.-G. Cai, S. Sun, B. Zhang, Y.-L. Zhang. Dark fluxes from accreting black holes through several mechanisms. arXiv:2009.02315.
  15. K. A. Malik, D. Wands. Cosmological perturbations. arXiv:0809.4944.

Final formulation

The proposed framework does not require the assumption that black holes themselves are dark matter.

It considers a weaker and testable possibility:

The BH population supplies a hypothetical dark source with a specified conversion efficiency.

with

\[\epsilon_X\ll1\]

and

\[\Omega_X<\Omega_{\rm DM}\]

The most important object of study is not an individual black hole, but the integrated historical throughput:

\[\mathcal T_X=\int Q_X(t)\left(\frac{a(t)}{a_0}\right)^3dt\]

Thus, the hypothesis is turned from a speculative claim into a problem of population synthesis + Boltzmann evolution + cosmological inference.

Its final status should be determined not by the energy estimate alone, but by the combined result:

BH demographics, accretion history, microscopic interactions, transport, CMB, large-scale structure and observations through multiple messengers.

With this formulation, the hypothesis can be turned into a quantitatively testable model of a subdominant dark-sector component.