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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 consider the phenomenological hypothesis that astrophysical black holes (BHs) may act as sources of a small, subdominant dark-sector component by converting part of the energy entering the horizon, or part of the energy released in the black-hole environment, into stable invisible degrees of freedom.

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=Q_X-3H(P_X-w_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\,da\,n_{\rm BH}(M,a,z)\,\Gamma_{\rm BH}(M,a,z)\,Y_X(M,a,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 key result is the separation of three fundamentally different quantities:

f_{\rm BH\rightarrow X} \equiv \frac{\dot{\rho}_{X,\rm inj}} {\dot{\rho}_{\rm BH,feed}}

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.

The main bottleneck of the hypothesis is not the availability of energy in BHs, but the need to satisfy simultaneously the energy budget, cosmological evolution, local constraints on the BH population, electromagnetic and gravitational-radiation constraints, and the kinematic requirements of the produced dark-sector component.

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)

where \(\mathcal S_X\) describes survival, redshifting, decay, and transport.

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, f_{\rm BH\rightarrow X}, f_{\rm bound}, f_{\rm surv}, m_X,\sigma_X,\ldots}

For a subdominant component, we require

\Omega_X < \Omega_{\rm DM}

with particular interest in the range

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

as a phenomenological benchmark, not as an established result.

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:

\text{ordinary matter} \rightarrow \text{BH} \rightarrow \text{dark sector}

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 {\rm few},M_\odot \rightarrow 10^9 M_\odot

they form from stars and grow through mergers and accretion, primarily within galactic structures.

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

{\rm BH\ environment} \rightarrow X

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

In the most general form:

\dot E_X = \epsilon_X \dot E_{\rm BH}

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

Black holes are extreme gravitational systems with:

– 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 can be much larger than the instantaneous energy output of a particular object.

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

4. Energy screening

The first question is whether enough energy is available at all.

Let

\rho_{\rm BH}(z)

be the comoving or physical BH mass density, depending on the chosen convention.

Then the available rest-mass scale is:

\rho_{\rm BH}c^2

For an accretion-driven channel, the accretion energy is:

E_{\rm acc}=\int dt\,\epsilon_{\rm acc}\dot M_{\rm in}c^2

with

\dot M_{\rm BH}=(1-\epsilon_{\rm acc})\dot M_{\rm in}

Therefore,

\dot E_{\rm rad}=\epsilon_{\rm acc}\dot M_{\rm in}c^2

Rather than identifying all of this energy with the dark-sector source, we introduce:

\dot E_X=f_{\rm conv}\dot E_{\rm available}

Here

f_{\rm conv}\ll1

This quantity must be calculated from a microscopic model.

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}=\frac{E_{X,\rm bound}}{E_{X,\rm produced}}

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.

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) d\rho_X(t’) \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\,da\,n_{\rm BH}(M,a,z) \Gamma_{\rm BH}(M,a,z)\,Y_X(M,a,z) M c^2

where:

– \(M\) is the BH mass;

– \(a\equiv J/(GM^2/c)\) — dimensionless spin;

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

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

– \(Y_X\) — yield dark-sector energy.

The last two parameters can be combined:

\eta_X(M,a,z)=\Gamma_{\rm BH}Y_X

Then

Q_X= c^2\int dM\,da\,n_{\rm BH} M\eta_X

7. Population-integrated throughput

The main physical quantity in this work is not instantaneous luminosity, but integrated throughput.

Define:

\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)

where

\mathcal S_X=f_{\rm surv}f_{\rm bound}

in the simplest approximation.

Then:

\rho_X(t)=\mathcal T_X(t)

For a stable cold component:

\rho_X(t)=\int_{t_{\rm form}}^{t} dt’\,Q_X(t’) \left[ \frac{a(t’)}{a(t)} \right]^3 f_{\rm bound}(t’,t)

This expression shows an important difference from the usual relic-density model.

The dark component may havehistorical memory.

That is, today’s \(\rho_X\) need not correlate with today’s BH activity.

8. Historical memory

Split the population into:

n_{\rm BH}(M,a,z)=n_{\rm seed}+n_{\rm growth}+n_{\rm merger}

Then:

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

For SMBHs, the accretion term is particularly natural:

Q_{\rm acc}\propto\int dM\,da\,n_{\rm BH}\,\dot M_{\rm acc}\,\eta_X

A merger-related contribution is also possible:

Q_{\rm merger} \propto \int dM_1\,dM_2\, R_{\rm merge} E_{\rm available} f_X

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\,da

9. Minimal benchmark model

To obtain testable scaling relations, introduce:

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

and

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

Then:

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

\alpha+\beta=0

then the source approximately traces the BH mass density:

Q_X\propto\rho_{\rm BH}

\alpha+\beta>0

then massive BHs dominate.

\alpha+\beta<0

then the contribution shifts toward smaller BHs.

This turns the model into a population-synthesis problem.

10. Black-hole mass function

Denote:

\Phi(M,z)=\frac{dn_{\rm BH}}{d\log M}

Then:

\rho_{\rm BH}(z)=\int d\log M\,M\Phi(M,z)

Source:

Q_X(z)=c^2\int d\log 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

If dark-sector production is sufficiently small, BH evolution is essentially unchanged.

In this case:

\dot M_{\rm BH}=\dot M_{\rm acc}+\dot M_{\rm merge}-\dot M_{\rm loss}

and \(X\)-production is a perturbation.

For the more general case:

\dot M_{\rm BH}=\dot M_{\rm standard}+\frac{Q_X}{c^2}

This gives the self-consistency condition:

\frac{Q_X} {\dot\rho_{\rm BH}c^2} \ll1

This criterion must hold for a subdominant source.

12. Main dimensionless ratio

The most informative parameter is:

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

\epsilon_X\ll1

then dark-sector production does not significantly alter the BH population.

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

A particularly interesting phenomenological range is:

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

This is not an observational determination, but a benchmark-model region.

14. Why PBHs must be considered separately

PBHs have a completely different temporal kernel.

For Hawking radiation:

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:

T_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:

{\rm accretion/rotation/environment} \rightarrow 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

{\rm accretion\ plasma} \rightarrow X+\cdots

if a weak coupling is present.

15.2. Gravitational production

T_{\mu\nu}^{\rm BH} \rightarrow X

through gravitationally suppressed interactions.

15.3. Horizon-associated channel

The process can be parameterized phenomenologically as:

{\rm horizon} \rightarrow X

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

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(a)

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\,da\,n_{\rm BH}(M,a,\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 the produced particles rapidly leave the 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 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.

18. Perturbations

At linear order:

\delta Q_X=Q_X\left(\delta_{\rm BH}+\delta_\Gamma+\delta_Y\right)

Thus, the source can create a correlation between dark-sector perturbations and the BH population.

In Fourier space:

\delta_X(k,z)=\int dz’\,G_X(k;z,z’)\delta Q_X(k,z’)

If the source is late, it does not fully reproduce the standard primordial cold-dark-matter transfer function.

This is a potentially important observational discriminator.

19. Effective bias

If the source is predominantly associated with AGN-host galaxies:

\delta Q_X \simeq b_{\rm BH}(z)\delta_m

Then the dark component acquires an effective bias:

b_X(k,z) \neq 1

on sufficiently large scales.

For a significant subcomponent:

\delta_m=(1-\xi)\delta_{\rm standard}+\xi\delta_X

Therefore:

\[P_m(k)\]

may differ from standard \(\Lambda\)CDM.

20. Early-Universe constraint

If the main source turns on too early, it may affect:

– BBN;

– recombination;

– CMB;

– matter-radiation equality.

Therefore, the model must satisfy:

\int^{z_{\rm CMB}} Q_X(z),dt \ll \rho_{\rm rad}(z_{\rm CMB})c^2

for a cold late-time production scenario.

This naturally motivates scenarios in which the source becomes significant after the formation of the first galaxies.

21. Late-time source

\[Q_X(z)\]

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:

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

Thus, even a small energy fraction can have a large structural impact if the produced component is warm/hot.

Therefore we require:

\lambda_{\rm FS}

to be smaller than the characteristic scale on which observational constraints are strongest.

23. Cold-production regime

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

\frac{p_{\rm prod}}{m_X} \ll1

or sufficiently strong redshifting before matter domination.

If particles are produced with

p_{\rm prod}\gg m_X

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

Introduce:

\rho_X=\rho_X^{\rm rel}+\rho_X^{\rm NR}

Then:

\dot\rho_X^{\rm rel}+4H\rho_X^{\rm rel}=Q_{\rm rel}

\dot\rho_X^{\rm NR}+3H\rho_X^{\rm NR}=Q_{\rm NR}

At the transition:

Q_{\rm rel}\rightarrow Q_{\rm NR}

This avoids the incorrect assumption that all injected energy instantaneously becomes cold matter.

25. Population synthesis

The practical calculation should be constructed as:

\Phi(M,a,z) \rightarrow \dot M(M,a,z) \rightarrow Q_X(M,a,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,a,z)=\Phi_M(M,z)P(a|M,z)

Then:

Q_X(z)=c^2\int d\log M\,M\Phi_M(M,z)\Gamma_X(M,z)\langle Y_X\rangle_a

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\log M\, M\Phi_M \left(\frac{M}{10^8M_\odot}\right)^\alpha

This is already a fully numerically implementable model.

27. Accretion-linked benchmark

A more physically transparent option is:

Q_X=f_X\epsilon_{\rm acc}\dot\rho_{\rm acc}c^2

Here:

\dot\rho_{\rm acc}=\int dM\,da\,n_{\rm BH}\dot M_{\rm acc}

Then:

Q_X=f_X\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.

28. Order-of-magnitude energy condition

The integrated produced density is:

\rho_Xc^2\simeq f_XE_{\rm acc,cosmic}

For a subdominant fraction \(\xi\):

\rho_X=\xi\rho_{\rm DM}

Therefore,

f_X \simeq \frac{\xi\rho_{\rm DM}c^2} {E_{\rm acc,cosmic}}

This is the main energy-screening equation.

If the required

\[f_X>1\]

the model is energetically excluded.

f_X\ll1

there is no energetic contradiction.

But this isnot proof of the mechanism.

29. What a small \(f_X\) means

If the calculation gives, for example,

f_X\sim10^{-6}

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

Such a result:

f_X\ll1

is not automatically evidence of physical realizability.

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

30. Null models

To test the hypothesis, it must be compared with several null models.

Null 1

\[Q_X=0\]

Standard \(\Lambda\)CDM.

Null 2

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

Null 3

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

Null 4

Q_X\propto\rho_{\rm BH}(z)\delta(z-z_*)

Null 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’) f_X \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)

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.

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)=\int dr’\,K(r,r’)Q_X(r’)

Therefore, central enhancement must be calculated, not assumed.

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\,dz\,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.

Current LVK analyses 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.

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:

\[f_X\]

cannot be considered independently of branching ratios.

37. CMB constraints

Late-time energy injection can change:

\[x_e(z)\]

the baryonic-gas temperature and CMB anisotropies.

In the case of decay:

Q_{\rm vis}=f_{\rm vis}Q_X

It is necessary to require:

Q_{\rm vis}(z) < Q_{\rm CMB}^{\rm max}(z)

Therefore, the most natural case is a sector with

f_{\rm vis}\ll1

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| < {\rm observational\ bound}

39. Halo abundance

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

Then:

f_X(z_{\rm form}) < f_X(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(r)=\sum_iQ_{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 dM\,M^{1+\alpha}\Phi_i(M)

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

42. Spin dependence

If:

\Gamma_X=\Gamma_0g(a)

then:

Q_X \propto \langle g(a)\rangle

For a superradiant mechanism:

\[g(a)\]

may increase sharply with spin.

Then the model gains an additional observable prediction:

\Omega_X=\Omega_X(M,a,z)

43. Information encoded in the BH population

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

{M,a,\dot M,z} \rightarrow Q_X

The current BH catalogue contains incomplete information about the past.

Therefore, reconstruction is required:

P(M,a,\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

Since:

L_{\rm AGN}=\epsilon_{\rm rad}\dot M_{\rm in}c^2

one can write:

Q_X(z)=f_X\int dL\,\Phi_{\rm AGN}(L,z)L

This is particularly useful because AGN luminosity functions are observable.

Thus:

Q_X(z) \leftrightarrow \Phi_{\rm AGN}(L,z)

is a potentially calibratable bridge.

46. Soltan-type consistency test

If:

\dot\rho_{\rm BH}=(1-\epsilon)\dot\rho_{\rm in}

then:

\dot\rho_{\rm in}=\frac{\dot\rho_{\rm BH}}{1-\epsilon}

Then:

Q_X=f_X\epsilon\frac{\dot\rho_{\rm BH}}{1-\epsilon}c^2

Therefore:

\frac{Q_X}{\dot\rho_{\rm BH}c^2}=f_X\frac{\epsilon}{1-\epsilon}

This is a convenient consistency relation.

Soltan-type analyses show that integrated AGN luminosity and local SMBH mass density are consistent in order of magnitude for standard radiative efficiencies, although the precise reconstruction of BH growth history has substantial systematics.

Recent JWST-era studies also indicate that early SMBH growth may contain a substantial obscured or radiatively inefficient component, underscoring the need for population synthesis rather than a single luminous-quasar population.

47. Fundamental limitation

Even if:

f_X\ll1

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

It is necessary to:

{\rm energy\ feasibility} \neq {\rm microphysical\ feasibility}

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

\[f_X>1\]

2. CMB violation

Q_{\rm vis}>Q_{\rm CMB}^{\rm max}

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,a,z)

that gives the required \(Q_X\).

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:

{\rm data} \Rightarrow Q_X(z) \neq0

and simultaneously:

\[Q_X(z)\]

must be statistically associated with

\Phi_{\rm BH}(M,a,z)

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

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

50. Statistical framework

Parameters:

\theta= { f_X,\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}}

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.

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_X \le 10^{-1}

-2\le\alpha\le2

-4\le\gamma\le4

and for the dark-sector particle:

10^{-6}\,{\rm eV} \le m_X \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.

54. Main expected regimes

Regime I — negligible

\xi\ll10^{-6}

Practically unobservable source.

Regime II — weak subcomponent

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

The most natural regime for weak coupling.

Regime III — appreciable subcomponent

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

Requires stronger observational tests.

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.

A late source naturally encounters the problem of:

{\rm timing}

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

PBH-DM:

{\rm early\ Universe} \rightarrow {\rm PBH} \rightarrow {\rm DM}

Astrophysical-BH source:

{\rm stars} \rightarrow {\rm galaxies} \rightarrow {\rm BH} \rightarrow X

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.

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:

{\rm astrophysical\ portal}

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

58. What is genuinely new in the framework

The novel part of the present approach is not the claim that BHs can emit or interact with the dark sector.

Such mechanisms have already been studied, especially for PBH Hawking radiation.

The main idea here is different:

{\rm astrophysical\ BH\ population} \rightarrow {\rm historical\ source} \rightarrow {\rm subdominant\ dark\ component}

with explicit inclusion of:

\Phi(M,a,z)

\dot M_{\rm acc}

f_{\rm conv}

f_{\rm bound}

f_{\rm surv}

and the transport kernel.

59. Strongest version of the hypothesis

The strongest formulation of the hypothesis is:

There exists a stable dark-sector component \(X\) whose energy density is an integral over the cosmological history of the astrophysical BH population, with the source term proportional to some function of black-hole mass, spin, and accretion history.

Mathematically:

\rho_X(t)=\int_{t_{\rm form}}^t dt’\int dM\,da\,n_{\rm BH}(M,a,t’)\,\Gamma_X(M,a,t’)\,Y_X(M,a,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. Most conservative 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 E_{\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:

{\rm energy\ bound} \rightarrow {\rm population\ bound} \rightarrow {\rm cosmological\ bound} \rightarrow {\rm transport\ bound} \rightarrow {\rm microphysics}

This prevents over-interpretation of the result.

62. Possible outcome of numerical analysis

Three situations may arise.

Case A

\[f_X>1\]

The hypothesis is energetically impossible.

Case B

10^{-10}<f_X<10^{-3}

The hypothesis is energetically viable and requires a microscopic mechanism for weak conversion.

Case C

f_X\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:

{\rm dark\ component} \leftrightarrow {\rm BH\ 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.

Thus:

{\rm correlation\ with\ BH/AGN\ history}

is the key potential distinguishing feature.

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.

Therefore, the present text is framework + hypothesis + falsifiable program, rather than a proof of the origin of dark matter.

67. What the paper claims

The paper claims only the following:

\text{Astrophysical BHs provide a physically large cosmic energy-throughput reservoir.}

and

\text{A sufficiently weak conversion channel could in principle generate a subdominant dark-sector component.}

The next test:

\text{Does any physically consistent microphysics realize the required conversion efficiency?}

remains an open question.

68. What the paper does NOT claim

The paper does not claim:

{\rm BH}=DM

It does not claim:

{\rm BH}=white\ holes

It does not claim:

{\rm BH}=all\ dark\ matter

It does not claim the existence of baby universes.

It does not claim confirmation of quantum gravity.

All of these ideas may be considered only as separate speculative extensions.

69. Speculative extensions

Possible models include:

BH\rightarrow white\ hole

BH\rightarrow baby\ universe

or other topology-changing processes.

But they require physics beyond the framework of this work.

Therefore, they are deliberately not used to obtain the main results.

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\,da\,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:

\text{the current BH population and historical BH throughput are different physical quantities.}

Therefore, testing the hypothesis should be based neither on individual black holes nor on today’s BH mass density, but on the integrated history:

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

Energetically, the scenario can be viable even with a very small conversion efficiency if the integrated BH energy throughput is sufficiently large. However, energetic viability is not evidence for the existence of a microscopic channel.

Therefore, the central scientific question is formulated as follows:

\begin{gathered} \mathrm{Can\ a\ physically\ consistent\ dark\!-\!sector\ interaction\ convert\ a\ sufficiently\ small\ fraction}\\ \mathrm{of\ the\ historical\ energy\ throughput\ of\ astrophysical\ black\ holes\ into\ a\ stable,}\\ \mathrm{gravitationally\ clustered\ component\ without\ violating\ cosmological,\ astrophysical,}\\ \mathrm{electromagnetic\ and\ gravitational\ constraints?} \end{gathered}

This question is concrete, quantitative, and potentially falsifiable.

Appendix A. Complete system of equations

For the cold component:

\dot\rho_X+3H\rho_X=Q_X

For the relativistic component:

\dot\rho_R+4H\rho_R=Q_R

For the BH population:

\frac{\partial n_{\rm BH}}{\partial t}+\frac{\partial}{\partial M}\left(\dot M_{\rm BH}n_{\rm BH}\right)=S_{\rm form}+S_{\rm merge}-D_{\rm merge}

With:

\dot M_{\rm BH}=(1-\epsilon_{\rm acc})\dot M_{\rm in}

Dark source:

Q_X=\int dM\,da\,n_{\rm BH}\dot M_{\rm in}c^2\epsilon_{\rm acc}f_X(M,a,z)

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.

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(a)

Appendix D. Benchmark-model parameter set

Minimal parameter set:

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

Additionally:

\beta

for a mass-dependent yield and

\[a_0\]

for spin dependence.

Appendix E. Main viability criterion

A scenario is phenomenologically viable only if all of the following hold simultaneously:

0<\xi<1

0<\epsilon_{\rm eff}<1

\Delta P(k)<\Delta P_{\rm max}

\Delta C_\ell<\Delta C_{\ell,\rm max}

Q_{\rm vis}<Q_{\rm vis,max}

and

\Delta\Phi_{\rm BH}

does not conflict with the observed BH population.

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, A&A 641, A6 (2020).

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, Effects of primordial black holes on dark matter models, Phys. Rev. D 102, 095018 (2020).

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, arXiv:0808.0070.

8. A. Marconi et al., Cosmological evolution of supermassive black holes, and subsequent Soltan-type population analyses. The mass-function/AGN-luminosity connection is reviewed in the BH-demography literature.

9. K. Jahnke, The Soltan argument at z=6: UV-luminous quasars contribute less than 10% to early black hole mass growth, Open Journal of Astrophysics (2025).

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, study of dark-sector production/detection associated with PBHs.

Final formulation

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

It considers a weaker and testable possibility:

\mathrm{BH\ population}\overset{\epsilon_X}{\longrightarrow}\mathrm{dark\ sector}

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:

\text{BH demographics}+\text{accretion history}+\text{microphysics}+\text{transport}+\text{CMB}+\text{LSS}+\text{multi-messenger constraints}

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