Pseudo-Mass Function: Concepts & Applications

Updated 10 July 2026

Pseudo-mass function is a family of constructions that redefine traditional mass by using evolving reference densities, constrained ensembles, and auxiliary dynamical parameters.
It is applied in diverse fields such as halo and cluster cosmology, PT-symmetric quantum mechanics, and modified statistical frameworks, impacting both simulation and observational interpretations.
The approach distinguishes between pseudo evolution (apparent mass change) and physical mass growth, providing critical insights for interpreting large-scale structure dynamics and non-Hermitian quantum systems.

“Pseudo-mass function” is a context-dependent term rather than a single standardized object. In cosmology it is used for apparent or environment-dependent halo and cluster mass functions produced by evolving reference densities, statistical constraints from large-scale structure, or modified collapse thresholds. In PT-symmetric quantum mechanics it refers to the role of the position-dependent effective mass $M(x)$ as a deforming element in $n$ -pseudo-Hermitian Hamiltonians. Related constructions also appear as a dynamically generated mass function $\Sigma(p,T)$ , a pseudo-mass parameter $\mu$ attached to nuclear-charge dynamics, and a pseudo-analytic mass density whose integral is an invariant (Shirasaki, 2019, Diemer et al., 2012, Kurbatov, 2011, Hiotelis et al., 2018, Soliemani et al., 2019, Fernández et al., 2020, Liu, 2020, Alayón-Solarz, 26 Jun 2026).

1. Terminological scope

The cited literature uses the term in several technically distinct ways. This suggests that “pseudo-mass function” is best understood as a family of constructions in which the operative mass variable is modified by a background definition, a similarity transformation, a constrained ensemble, or an auxiliary dynamical extension.

Domain	Object called “pseudo-mass” or “pseudo-mass function”	Defining feature
Halo and cluster cosmology	SO mass evolution or constrained halo mass function	Depends on reference density or LSS constraints
Excursion-set theory	First-crossing mass function with modified barrier	Uses a lower than usual constant collapse threshold
PT-symmetric quantum mechanics	Position-dependent effective mass $M(x)$	Deforms kinetic operator and pseudo-Hermitian structure
PQED + Gross-Neveu theory	$\Sigma(p,T)$	Dynamically generated interaction-induced mass
Alchemical quantum theory	Pseudo-mass parameter $\mu$	Gives a kinetic term in nuclear-charge space
Beltrami–Vekua theory	Pseudo-analytic mass density $\Theta$	Invariant 2-form whose integral is total mass

A common structural feature is that the mass variable is not treated as a bare, directly measured quantity. Instead, it is redefined by a reference convention, constrained by a background field, or introduced as an auxiliary degree of freedom.

2. Pseudo-evolution in halo and cluster mass functions

In the spherical-overdensity framework, halo mass is defined by

$M_{\Delta}(z)=\frac{4}{3}\pi R_\Delta^3(z)\,\Delta(z)\rho_{\rm ref}(z),$

with the enclosed-mass representation

$M_{\Delta}(z)=\int_0^{R_\Delta(z)}4\pi r^2dr\,\rho(r,z).$

The redshift derivative can be decomposed as

$n$ 0

In this decomposition, the first term is pseudo evolution and the second term is physical mass growth. Pseudo evolution is the apparent change in SO mass caused by the redshift evolution of the reference density even when the physical density profile remains fixed in physical coordinates (Shirasaki, 2019, Diemer et al., 2012).

The simulation-based quantification is explicit. For clusters with virial mass $n$ 1 at $n$ 2, the mean fraction of the SO-mass difference between $n$ 3 and $n$ 4 attributed to pseudo evolution is

$n$ 5

The increment between adjacent snapshots is written as

$n$ 6

and the cumulative fraction as

$n$ 7

For galaxy-scale halos with $n$ 8, pseudo evolution accounts for nearly all measured mass growth from $n$ 9 to $\Sigma(p,T)$ 0, while for cluster-scale halos with $\Sigma(p,T)$ 1 it accounts for at least one third to over half of the measured mass growth (Shirasaki, 2019, Diemer et al., 2012).

The mass-function consequences are substantial. SO-defined masses grow even without physical accretion, which affects inferences from cluster counts, scaling relations, and mass accretion histories. The non-evolution of the low-mass end of the halo mass function is described as a fortuitous cancellation between pseudo-evolution and the absorption of small halos into larger hosts. The observed evolution of the low-mass end of the concentration–mass relation is described as almost entirely due to pseudo-evolution of mass (Diemer et al., 2012).

3. Environment-conditioned and observation-conditioned statistical mass functions

A different use of “pseudo-mass function” appears in modified Press–Schechter theory with a background large-scale structure. The overdensity field is constrained by linear functionals

$\Sigma(p,T)$ 2

and decomposed into a constrained mean and residual field,

$\Sigma(p,T)$ 3

The residual variance becomes

$\Sigma(p,T)$ 4

so the mass function depends on position relative to the host supercluster or void. The resulting constrained mass function is

$\Sigma(p,T)$ 5

and is explicitly interpreted in a pseudo-cosmological sense: the local background density defines its own effective cosmological parameters and growth history (Kurbatov, 2011).

In excursion-set theory, “pseudo-mass function” denotes a first-crossing mass function obtained from a stochastic process

$\Sigma(p,T)$ 6

with kernel

$\Sigma(p,T)$ 7

The analysis compares Monte Carlo first-crossing distributions with analytical approximations and finds that a constant threshold of collapse lower than the usual spherical-collapse value yields good agreement with $\Sigma(p,T)$ 8-body simulations. The modified barrier is

$\Sigma(p,T)$ 9

and the corresponding first-crossing distribution is described as the pseudo-mass function (Hiotelis et al., 2018).

Observation-conditioned usages appear in both cluster and exoplanet statistics. In the REFLEX II analysis, a “pseudo-mass function construction” converts the X-ray luminosity function to a mass function through the scaling relation $\mu$ 0, but the preferred procedure is not to fit that pseudo-mass function directly: the model X-ray luminosity function derived from the theoretical mass function and scaling relation is fitted to the observed luminosity distribution, and the mass function corresponding to the best-fit cosmological parameters is then adopted (Boehringer et al., 2017). In exoplanet statistics, the planetary mass function is explicitly not treated as a low-mass extension of the stellar mass function. Instead, the fundamental object is the differential planetary mass-radius-orbit function

$\mu$ 1

with the host-conditioned marginal

$\mu$ 2

and the population-integrated function $\mu$ 3. In that account, the observed planetary mass function is a pseudo-mass function because it is convolved over host-star properties, multiplicity, selection biases, and detection efficiencies (Dominik, 2010).

4. Position-dependent effective mass as pseudo-mass in PT-symmetric quantum mechanics

In one-dimensional non-Hermitian quantum systems with PT symmetry, the position-dependent effective mass $\mu$ 4 is treated as the pseudo-mass. The starting point is a Dirac equation with position-dependent effective mass in an external field, and after reduction one obtains a Schrödinger-like operator with kinetic term

$\mu$ 5

The pseudo-Hermitian construction uses two first-order operators,

$\mu$ 6

with $\mu$ 7 in the procedure, together with the generalized relation

$\mu$ 8

The mass function enters the kinetic operator and the effective potential, and thus participates directly in the metric structure associated with $\mu$ 9-pseudo-hermiticity (Soliemani et al., 2019).

A canonical transformation rewrites the problem in terms of a new coordinate $M(x)$ 0 satisfying

$M(x)$ 1

so that the reduced equation becomes

$M(x)$ 2

with

$M(x)$ 3

This formulation allows the use of known solution techniques for standard solvable models, while the mass function is encoded through the transformation $M(x)$ 4 (Soliemani et al., 2019).

The formalism is applied to complex Pöschl–Teller and Eckart potentials. For the Pöschl–Teller case,

$M(x)$ 5

with pseudo-Hermitic operator input

$M(x)$ 6

For the Eckart potential,

$M(x)$ 7

The paper states that suitably chosen mass functions and intertwining operators can yield non-Hermitian Hamiltonians with real spectra, and that the spectra can display level crossing phenomena (Soliemani et al., 2019).

5. Dynamical pseudo-mass and pseudo-mass parameters in extended quantum dynamics

In Pseudo Quantum Electrodynamics coupled to the Gross–Neveu interaction, the pseudo-mass function is the dynamically generated fermion mass $M(x)$ 8. The dressed fermion propagator is written as

$M(x)$ 9

and the Schwinger–Dyson equation separates the contributions from the long-range PQED interaction and the Gross–Neveu channel,

$\Sigma(p,T)$ 0

The mass is “pseudo” in the sense that it is not a bare parameter but emerges dynamically when the coupling is strong enough or the number of fermion species is low enough. In the static regime the analysis yields a critical number of fermions

$\Sigma(p,T)$ 1

and, at finite temperature,

$\Sigma(p,T)$ 2

The critical coupling $\Sigma(p,T)$ 3 is also given explicitly, and finite temperature suppresses mass generation (Fernández et al., 2020).

In the pseudo-mass parameterized alchemical equation, the pseudo-mass is instead an auxiliary parameter $\Sigma(p,T)$ 4 attached to a kinetic term in nuclear-charge space: $\Sigma(p,T)$ 5 This formulation treats nuclear charges, nuclear coordinates, and electronic coordinates on an equal footing. The total wave function is factorized as

$\Sigma(p,T)$ 6

where $\Sigma(p,T)$ 7 is the nuclear-charge wave function. In the limit $\Sigma(p,T)$ 8, nuclear charge becomes a frozen parameter and conventional quantum chemistry is recovered (Liu, 2020).

The same framework defines an alchemical function space

$\Sigma(p,T)$ 9

derives an alchemical phase space with

$\mu$ 0

and obtains, for a hydrogen-like ion, the inverse-square equation

$\mu$ 1

The construction also includes a geometric phase and an extended Hellmann–Feynman theorem for non-stationary, time-dependent clamped-nuclear-charge dynamics (Liu, 2020).

6. Geometric and analytic generalizations

In the framed Beltrami–Vekua normal form, the relevant object is a pseudo-analytic mass density 2-form rather than a mass function in the cosmological sense. For the equation

$\mu$ 2

the density is

$\mu$ 3

The total mass

$\mu$ 4

is invariant under recombination of unknowns and under scaling, and is covariant under orientation-preserving changes of variables. At the trivial frame $\mu$ 5, this reduces to

$\mu$ 6

The paper identifies this with the pseudo-analytic mass density of the unframed equation (Alayón-Solarz, 26 Jun 2026).

In compact-star modeling, a different analytic construction appears. The proposed specific mass function is

$\mu$ 7

with associated metric component

$\mu$ 8

The solution is constructed in embedding class one spacetime, and the summary states that the pseudo-mass function naturally arises due to this embedding. The model is regular at the center, satisfies the energy conditions, obeys Buchdahl’s bound, and is compared with objects including 4U1608-52, PSR J1903+327, PSR J1614-2230, and Vela X-1 (Maurya et al., 2017).

A still more geometric usage appears in the folded pseudo-coordinate approach to mass. There, pseudo-mass is the residual observable real mass that emerges from folding the fourth coordinate with symmetry

$\mu$ 9

The formulation distinguishes real mass densities from imaginary electromagnetic mass densities, relates the Ricci scalar curvature to active and passive real mass densities, and treats electromagnetic source and sink flows as anti-collinear before folding and collinear after folding. This yields a pseudo-mass as a residual, real mass left over after imaginary mass densities are geometrically folded in the pseudo-geometry (Bulyzhenkov, 2008).

Taken together, these usages show that the pseudo-mass function is not a single formula but a recurring strategy: redefine mass through a changing reference density, a constrained stochastic ensemble, a non-Hermitian operator algebra, or an auxiliary geometric extension. The term therefore marks a modification of the mass concept itself, rather than a unique formalism shared across fields.