Quadratic EOS: Stellar & Cosmological Applications

• Quadratic equation of state is a nonlinear constitutive law relating pressure to density via a quadratic term, capturing higher-order interaction effects in dense matter.
• It is employed in modeling anisotropic compact stars, dark energy fluids, and unified dark fluids, offering analytical tractability and effective parameterizations.
• Its versatility extends to modified gravity scenarios and freshwater convection, enabling nuanced control of stability, causality, and density-dependent stiffness.

A quadratic equation of state (EOS) is a constitutive relation in which pressure, or an analogous thermodynamic variable, depends quadratically on density or on another state variable. In relativistic stellar modeling, the canonical form is a barotropic law such as $p_r=\gamma\rho^2+\alpha\rho-\beta$ or a closely related variant; in cosmology, the term is also used for dark-energy fluids of the form $p=w\rho+\beta\rho^2$, for redshift-parameterized EOS parameters such as $\omega(z)=-1+\alpha(1+z)+\beta(1+z)^2$, and in fluid dynamics for quadratic density–temperature laws near a density maximum (Ratanpal et al., 22 Aug 2025, Moshafi et al., 2024, Bhardwaj et al., 15 Apr 2025, Olsthoorn et al., 2020).

1. Formal definitions and major variants

The phrase “quadratic equation of state” is not attached to a single universal formula. In compact-star theory it usually denotes a nonlinear barotropic law relating pressure and energy density, whereas in some cosmological papers it denotes either a nonlinear pressure law or a quadratic parameterization of the EOS parameter itself. A further variant appears in convection problems, where density depends quadratically on temperature rather than pressure depending on density.

Context	Representative form	Role
Anisotropic compact stars	$p_r=\gamma\rho^2+\alpha\rho-\beta$ or $p_r=(1-r^2/R^2)[A\rho^2+B\rho]$	Closes Einstein or Einstein–Maxwell system (Maharaj et al., 2013, Ratanpal et al., 22 Aug 2025)
Dark-energy fluid	$$p_{\rm v}=w_{\rm v}\rho_{\rm v}+\beta\rho_{\rm v}^2$$	Gives density-dependent $w(z)$ and $c_s^2(z)$ (Moshafi et al., 2024)
Modified-gravity EOS parameterization	$\omega(z)=-1+\alpha(1+z)+\beta(1+z)^2$	Phenomenological redshift law (Bhardwaj et al., 15 Apr 2025)
Unified dark fluid	$$p_g=p_0\rho_{\rm cr}+w_0\rho_g+\beta\rho_g^2/\rho_{\rm cr}$$	Single-fluid alternative to $p=w\rho+\beta\rho^2$0CDM (Sharov, 2015)
Freshwater near density maximum	$p=w\rho+\beta\rho^2$1	Captures density maximum near $p=w\rho+\beta\rho^2$2C (Olsthoorn et al., 2020)

The standard compact-star interpretation emphasizes the additional $p=w\rho+\beta\rho^2$3 term as a phenomenological representation of non-linear interaction effects between baryons or quarks, higher-order corrections to nuclear or quark-matter EOS, and strong-field or strong-coupling contributions. In that setting, the effective stiffness becomes density dependent because $p=w\rho+\beta\rho^2$4, rather than constant as in a linear EOS (Ratanpal et al., 22 Aug 2025).

A distinct but related usage appears in exact-solution studies where the quadratic law is not imposed a priori. In the Finch–Skea anisotropic model of Sharma and Ratanpal, the exact $p=w\rho+\beta\rho^2$5 relation was obtained parametrically from the field equations and then fitted by $p=w\rho+\beta\rho^2$6; for their representative case VI with $p=w\rho+\beta\rho^2$7 km and $p=w\rho+\beta\rho^2$8, the fit $p=w\rho+\beta\rho^2$9 was reported to be almost identical to the exact parametric curve (Sharma et al., 2013).

2. Role in relativistic stellar structure

In static spherically symmetric compact-star models, the Einstein or Einstein–Maxwell equations are typically underdetermined unless additional physics is supplied through an EOS or a metric ansatz. Several exact constructions therefore use a quadratic EOS for the radial pressure $\omega(z)=-1+\alpha(1+z)+\beta(1+z)^2$0, while tangential pressure $\omega(z)=-1+\alpha(1+z)+\beta(1+z)^2$1 is determined from anisotropy and the field equations rather than by an independent EOS (Maharaj et al., 2013).

Maharaj and Mafa Takisa formulated charged anisotropic Einstein–Maxwell solutions with

$\omega(z)=-1+\alpha(1+z)+\beta(1+z)^2$2

combined with the Durgapal–Bannerji variables $\omega(z)=-1+\alpha(1+z)+\beta(1+z)^2$3, $\omega(z)=-1+\alpha(1+z)+\beta(1+z)^2$4, and $\omega(z)=-1+\alpha(1+z)+\beta(1+z)^2$5. With the ansätze

$\omega(z)=-1+\alpha(1+z)+\beta(1+z)^2$6

they obtained elementary-function solutions and showed that earlier linear- and quadratic-EOS models are recovered as special subfamilies (Maharaj et al., 2013).

Malaver used the Feroze–Siddiqui quadratic law

$\omega(z)=-1+\alpha(1+z)+\beta(1+z)^2$7

for uncharged anisotropic strange-quark-star configurations, together with the potential $\omega(z)=-1+\alpha(1+z)+\beta(1+z)^2$8 for $\omega(z)=-1+\alpha(1+z)+\beta(1+z)^2$9. In that construction, the case $p_r=\gamma\rho^2+\alpha\rho-\beta$0 reduces to $p_r=\gamma\rho^2+\alpha\rho-\beta$1, which was identified with the polytropic case relevant to the Thirukkanesh–Ragel solutions (Malaver, 2014).

Recent uncharged models retain the same basic quadratic structure but embed it in more specialized geometries. In a paraboloidal interior, the radial metric potential is fixed by

$p_r=\gamma\rho^2+\alpha\rho-\beta$2

and the EOS becomes

$p_r=\gamma\rho^2+\alpha\rho-\beta$3

so the coefficients themselves vanish at the surface and enforce $p_r=\gamma\rho^2+\alpha\rho-\beta$4 automatically (Ratanpal et al., 22 Aug 2025). In the generalized Tolman–Kuchowicz spacetime, the metric

$p_r=\gamma\rho^2+\alpha\rho-\beta$5

is combined with

$p_r=\gamma\rho^2+\alpha\rho-\beta$6

and the matching conditions determine $p_r=\gamma\rho^2+\alpha\rho-\beta$7 for an observed mass–radius pair (Acharya et al., 3 Apr 2025).

A parallel charged, anisotropic construction in isotropic coordinates uses

$p_r=\gamma\rho^2+\alpha\rho-\beta$8

with $p_r=\gamma\rho^2+\alpha\rho-\beta$9 and $p_r=(1-r^2/R^2)[A\rho^2+B\rho]$0. In that framework, the linear case is recovered when $p_r=(1-r^2/R^2)[A\rho^2+B\rho]$1, and explicit stellar models were produced for PSR J1614-2230, 4U 1608-52, PSR J1903+0327, EXO 1745-248, and SAX J1808.4-3658 (Ngubelanga et al., 2015).

3. Regularity, causality, and stability in compact-star models

Quadratic-EOS stellar models are usually assessed by the standard criteria for realistic compact objects: regular metric functions at the center, finite and positive central density and pressures, monotonic decrease of $p_r=(1-r^2/R^2)[A\rho^2+B\rho]$2, $p_r=(1-r^2/R^2)[A\rho^2+B\rho]$3, and often $p_r=(1-r^2/R^2)[A\rho^2+B\rho]$4, vanishing $p_r=(1-r^2/R^2)[A\rho^2+B\rho]$5 at the surface, subluminal sound speeds, energy conditions, and dynamical stability indicators such as the adiabatic index and cracking conditions (Ratanpal et al., 22 Aug 2025).

In the paraboloidal model, matching to the Schwarzschild exterior imposes

$p_r=(1-r^2/R^2)[A\rho^2+B\rho]$6

hence $p_r=(1-r^2/R^2)[A\rho^2+B\rho]$7 and compactness $p_r=(1-r^2/R^2)[A\rho^2+B\rho]$8. The same model is reported to be non-singular, to satisfy NEC, WEC, and SEC throughout the star, and to preserve causality through

$p_r=(1-r^2/R^2)[A\rho^2+B\rho]$9

Its surface redshift is

$$p_{\rm v}=w_{\rm v}\rho_{\rm v}+\beta\rho_{\rm v}^2$$0

while the interior redshift is stated to remain bounded by $$p_{\rm v}=w_{\rm v}\rho_{\rm v}+\beta\rho_{\rm v}^2$$1 (Ratanpal et al., 22 Aug 2025).

The charged Einstein–Maxwell class of Maharaj and Mafa Takisa was designed partly to remove an earlier pathology: the proper charge density $$p_{\rm v}=w_{\rm v}\rho_{\rm v}+\beta\rho_{\rm v}^2$$2 may be singular at the center unless the electromagnetic parameter $$p_{\rm v}=w_{\rm v}\rho_{\rm v}+\beta\rho_{\rm v}^2$$3 is chosen appropriately. For $$p_{\rm v}=w_{\rm v}\rho_{\rm v}+\beta\rho_{\rm v}^2$$4, the paper gives

$$p_{\rm v}=w_{\rm v}\rho_{\rm v}+\beta\rho_{\rm v}^2$$5

so $$p_{\rm v}=w_{\rm v}\rho_{\rm v}+\beta\rho_{\rm v}^2$$6, and the central singularity in the proper charge density disappears (Maharaj et al., 2013).

The GTK model for PSR J0952–0607 combines the quadratic EOS with the Tolman–Oppenheimer–Volkoff equation, the Harrison–Zeldovich–Novikov criterion, Buchdahl’s bound, causality, the adiabatic index, and Herrera’s cracking condition. For the specific choice $$p_{\rm v}=w_{\rm v}\rho_{\rm v}+\beta\rho_{\rm v}^2$$7, $$p_{\rm v}=w_{\rm v}\rho_{\rm v}+\beta\rho_{\rm v}^2$$8, the paper reports a stability window

$$p_{\rm v}=w_{\rm v}\rho_{\rm v}+\beta\rho_{\rm v}^2$$9

because smaller $w(z)$0 gives cracking-type instabilities near the center, whereas larger $w(z)$1 leads to superluminal transverse sound speed in some region (Acharya et al., 3 Apr 2025).

In the isotropic charged model, the authors state that the PSR J1903+0327 configuration is physically reasonable, with regular metric potentials, positive central density and pressure, vanishing central electric field, and a smooth increase of the mass function to the observed mass $w(z)$2 at the boundary (Ngubelanga et al., 2015).

4. Dark energy, unified fluids, and observational cosmology

A major cosmological use of quadratic EOS ideas is the replacement of a cosmological constant by a nonlinear barotropic fluid. In one late-time dark-energy model,

$w(z)$3

with $w(z)$4. This gives an effective EOS parameter and adiabatic sound speed

$w(z)$5

so both vary with redshift even when $w(z)$6 is constant (Moshafi et al., 2024).

That model was implemented in CAMB and CosmoMC and constrained with Planck 2018 CMB data, Pantheon supernovae, BAO, and the SH0ES $w(z)$7 prior R21. The new parameter $w(z)$8 was reported to be effectively unconstrained for most data combinations; only CMB+SN+R21 yielded a meaningful upper bound, $w(z)$9. Bayesian evidence showed a decisive preference for the quadratic-EOS model in the restricted CMB+R21 comparison, with $c_s^2(z)$0, but a borderline strong preference for $c_s^2(z)$1CDM for the full CBS combination, with $c_s^2(z)$2. The same study concluded that the model does not fully resolve either the $c_s^2(z)$3 tension or the $c_s^2(z)$4 tension once all data are combined (Moshafi et al., 2024).

A related unified-dark-fluid model adopts

$c_s^2(z)$5

and fits low-redshift supernova, BAO, and $c_s^2(z)$6 data. In that comparison, the quadratic-EOS model attained the lowest raw $c_s^2(z)$7 among the models tested,

$c_s^2(z)$8

but it was not favored by the Akaike information criterion because it carries two additional parameters relative to $c_s^2(z)$9CDM (Sharov, 2015).

Quadratic structure has also been assigned to dark matter rather than dark energy: $\omega(z)=-1+\alpha(1+z)+\beta(1+z)^2$0 In that formulation, the background expansion and linear structure growth both change because

$\omega(z)=-1+\alpha(1+z)+\beta(1+z)^2$1

and the authors emphasize a distinct imprint on $\omega(z)=-1+\alpha(1+z)+\beta(1+z)^2$2, with potential relevance to the $\omega(z)=-1+\alpha(1+z)+\beta(1+z)^2$3 and JWST massive-galaxy tensions, while stating that the model does not directly solve the $\omega(z)=-1+\alpha(1+z)+\beta(1+z)^2$4 tension (Rezazadeh et al., 14 Sep 2025).

5. Inflation, bounces, and modified-gravity realizations

Quadratic EOS forms are also used to construct non-singular or bouncing cosmologies. A particularly influential example is

$\omega(z)=-1+\alpha(1+z)+\beta(1+z)^2$5

which was presented as a single analytic model of early inflation, intermediate decelerating expansion, and late accelerating expansion. In that framework, $\omega(z)=-1+\alpha(1+z)+\beta(1+z)^2$6 and $\omega(z)=-1+\alpha(1+z)+\beta(1+z)^2$7 act as upper and lower density bounds, the early and late universe are described by polytropic indices $\omega(z)=-1+\alpha(1+z)+\beta(1+z)^2$8 and $\omega(z)=-1+\alpha(1+z)+\beta(1+z)^2$9, and the model is stated to be free of a singularity at $$p_g=p_0\rho_{\rm cr}+w_0\rho_g+\beta\rho_g^2/\rho_{\rm cr}$$0 and to exist eternally in the past (Chavanis, 2013).

A one-parameter specialization,

$$p_g=p_0\rho_{\rm cr}+w_0\rho_g+\beta\rho_g^2/\rho_{\rm cr}$$1

was confronted with supernovae and BAO and found to fit as well as $$p_g=p_0\rho_{\rm cr}+w_0\rho_g+\beta\rho_g^2/\rho_{\rm cr}$$2CDM. The corresponding universe has no initial singularity but a “mild bounce,” and in the hot-bounce branch the bounce temperature is reported to lie above the Planck temperature (Berteaud et al., 2018).

Nonlinear electrodynamics provides another route to the same structure. In a magnetic FLRW universe, the Lagrangian

$$p_g=p_0\rho_{\rm cr}+w_0\rho_g+\beta\rho_g^2/\rho_{\rm cr}$$3

yields an effective quadratic EOS for generalized radiation,

$$p_g=p_0\rho_{\rm cr}+w_0\rho_g+\beta\rho_g^2/\rho_{\rm cr}$$4

in the early regime, and the author argues that this framework can simultaneously describe early inflation, a radiation era, and a dark-energy era without an initial singularity (Chavanis, 2024).

Modified gravity papers use the same language in two further ways. In Bianchi type-I $$p_g=p_0\rho_{\rm cr}+w_0\rho_g+\beta\rho_g^2/\rho_{\rm cr}$$5 cosmology, the choice

$$p_g=p_0\rho_{\rm cr}+w_0\rho_g+\beta\rho_g^2/\rho_{\rm cr}$$6

is combined with two classes of $$p_g=p_0\rho_{\rm cr}+w_0\rho_g+\beta\rho_g^2/\rho_{\rm cr}$$7 action and with power-law or exponential volume expansion; the resulting models show decreasing energy density, negative pressure, and late-time isotropization as the shear scalar decays (Singh et al., 2015). In $$p_g=p_0\rho_{\rm cr}+w_0\rho_g+\beta\rho_g^2/\rho_{\rm cr}$$8 gravity, the phrase “quadratic EOS” is applied to the redshift law

$$p_g=p_0\rho_{\rm cr}+w_0\rho_g+\beta\rho_g^2/\rho_{\rm cr}$$9

which, under joint CC+Pantheon+BAO analysis, gives

$p=w\rho+\beta\rho^2$00

with late-time SEC violation but WEC, NEC, and DEC satisfied (Bhardwaj et al., 15 Apr 2025).

6. Interpretation, effective descriptions, and cross-disciplinary analogues

Across stellar and cosmological applications, the quadratic term is usually interpreted phenomenologically rather than microscopically. In compact stars it is used to represent non-linear density effects in ultra-dense matter, to stiffen or soften the EOS depending on parameter signs, and to keep exact solutions analytically tractable in otherwise underdetermined relativistic systems (Ratanpal et al., 22 Aug 2025).

This effective viewpoint also appears when the quadratic law is fitted a posteriori rather than imposed. In the Finch–Skea anisotropic model, the exact $p=w\rho+\beta\rho^2$01 curve generated by a chosen pressure profile was found to be almost indistinguishable from a quadratic polynomial in $p=w\rho+\beta\rho^2$02, which indicates that quadratic forms can serve as compact surrogates for more complicated parametric relations over the density range sampled inside a star (Sharma et al., 2013).

The same constitutive idea extends beyond astrophysical pressure–density laws. In the “cooling-box” problem for freshwater, the quadratic EOS

$p=w\rho+\beta\rho^2$03

captures the density maximum near $p=w\rho+\beta\rho^2$04C. Three-dimensional direct numerical simulations showed that, under this nonlinear EOS, vertical heat transport is fundamentally different from the linear-EOS prediction and substantially lower, and the authors introduced an effective Rayleigh number

$p=w\rho+\beta\rho^2$05

as the analogue of the standard linear-EOS Rayleigh number (Olsthoorn et al., 2020).

Taken together, these usages suggest that “quadratic equation of state” denotes a family of nonlinear constitutive laws rather than a single microphysical theory. What unifies the family is not a unique ontology but a common mathematical feature: a second-order dependence that introduces density- or state-dependent stiffness, sound speed, and stability properties, while remaining simple enough to admit exact solutions, controlled approximations, or direct confrontation with data.