The spatial volume, \(V = a^3 = ABC\)
Hubble parameter, \(H = \frac{\dot{a}}{a}\)
The expansion scalar \(\theta\) = 3H
shear scalar, \(\sigma^2 = = \frac{1}{2} \left( H^2_x + H^2_y + H^2_z \right) – \frac{\theta^2}{6}\)
Anisotropy parameter \(A_m = \frac{2 \sigma^2}{3H^2}\)
Deceleration parameter, \(q = – \frac{\ddot{a}a}{\dot{a}^2}\) = \(\frac{1}{dt}\left(\frac{1}{H}\right) – 1\) = \(-\frac{1}{H^2}\frac{\ddot{a}}{a}\)
We can find different models of dark energy according to the values of the couple \((r, s)\). In particular,
The state finder pair {r, s} is define as
\(r = \frac{\dddot{a}}{aH^3}\)
\(s= \frac{r-1}{3(1-\frac{1}{2})}\)
CDM corresponds to \( (r = 1,\ s = 0)\);
SCDM corresponds to \( (r = 1,\ s = 1)\);
HDE corresponds to \( r = 1,\ s = \frac{2}{3}\);
CG corresponds to \( (r > 1,\ s < 0)\);
Quintessence corresponds to \( (r < 1,\ s > 0)\).
Source : link “https://www.mdpi.com/2218-1997/7/10/362“
jerk parameter, \( j = \frac{1}{H^3}\frac{\dddot{a}}{a}\)
snap parameter, \( s = \frac{1}{H^4}\frac{\ddddot{a}}{a}\)
Pressure P =
Energy Density \(\rho\) =
The equation of state (EoS) parameter is an important tool for comprehending the history of the Universe. If \(\omega = – 1\), it represents the CDM model; \(-1 < \omega < -\frac{1}{3}\) represents the quintessence model; and \(\omega < -1\) indicates the phantom behavior of the model.
EoS \(\omega = \frac{p}{\rho}\)
| Component | \(\omega\) | Description |
|---|---|---|
| Cosmological Constant (Λ) | \(\omega = -1\) | Dark energy with constant density |
| Quintessence | \(-1 < \omega < -\frac{1}{3}\) | Dynamic dark energy |
| Matter (dust) | \(\omega = 0\) | Pressureless matter (CDM, baryons) |
| Radiation | \(\omega = \frac{1}{3}\) | Relativistic particles (e.g., photons) |
| Stiff Fluid | \(\omega = 1\) | Hypothetical case |
| Phantom Energy | \(\omega < -1\) | Exotic dark energy with unusual behavior |
The evolution of energy conditions define as:
\(\text{WEC:} \quad \rho \geq 0 \)
\(\text{NEC:} \quad \rho + p \geq 0 \)
\(\text{DEC:} \quad \rho – p \geq 0\)
\(\text{SEC:} \quad \rho + 3p \geq 0\)
Weak Energy Condition \((WEC): \rho \geq 0\)
- Interpretation: Energy density is non-negative for any observer.
- Physical meaning: Matter/energy cannot have negative density — all observers must measure positive energy.
The Weak Energy Condition (WEC) is a concept from general relativity that places a constraint on the energy-momentum tensor, which describes the distribution and flow of energy and momentum in spacetime.
Null Energy Condition \((NEC): \rho + p \geq 0\)
- Interpretation: Energy density measured along a light-like (null) path is non-negative.
- Physical meaning: Prevents faster-than-light effects and helps preserve causality.
The Null Energy Condition says that the energy you measure along the path of light can’t be negative. It means that, even for light, energy should always be zero or positive, which helps keep physics behaving normally in the universe.
Dominant Energy Condition \((DEC): \rho – p \geq 0\)
- Interpretation: Energy flow is causal (not faster than light), and energy dominates over pressure.
- Physical meaning: Ensures matter moves slower than light and energy doesn’t flow in spacelike directions.
The Dominant Energy Condition means that the energy density is always positive, and energy never flows faster than light. In other words, the energy you measure is not only positive but also moves in a physically possible way, respecting causality.
Strong Energy Condition \((SEC): \rho + 3p \geq 0\)
- Interpretation: Gravity is attractive.
- Physical meaning: Implies decelerating expansion (in classical GR) and affects the convergence of geodesics.
The Strong Energy Condition says that gravity should always be attractive, meaning the combined energy and pressure in matter cause spacetime to focus or pull things together, not push them apart. It ensures the universe behaves in a way that leads to gravity’s usual “pull.”
Calculation of the variation Formula
\(Q_{\alpha \mu \nu}\) = \(\nabla_{\alpha} g_{\mu \nu}\)
\(Q^{\alpha \mu \nu}\) = \(-\nabla^{\alpha} g^{\mu \nu}\) = \(- \, g^{\alpha \rho} \nabla_{\rho} g^{\mu \nu}\)
\(Q^{\alpha}_{\ \mu \nu}\) = \(g^{\alpha \beta} Q_{\beta \mu \nu} \) = \(g^{\alpha \beta} \nabla_{\beta} g_{\mu \nu}\) = \(\nabla^{\alpha} g_{\mu \nu}\)
\(Q_{\alpha}{}^{\ \mu}{}_{\ \nu}\) = \( -g_{\rho \nu} \nabla_{\alpha} g^{\mu \rho}\)
\(Q_{\alpha \mu}{}^{ \ \nu}\) = \(-g_{\mu \rho} \nabla_{\alpha} g^{\nu \rho}\)
\(Q^{\alpha \mu}{}_{ \ \nu}\) = \(-g_{\rho \nu} \nabla^{\alpha} g^{\mu \rho}\)
\(Q^{\alpha \ \ \nu}_{\ \mu}\) = \(g^{\alpha \beta} g^{\nu \rho} \nabla_{\beta} g_{\mu \rho}\) = \(g^{\nu \rho} \nabla^{\alpha} g_{\mu \rho}\) = \(-g_{\mu \rho} \nabla^{\alpha} g^{\nu \rho}\)
\(Q_{ \alpha}^{\ \ \mu \nu}\) = \(g^{\mu \rho} g^{\nu \sigma} \nabla_{\alpha} g_{\rho \sigma}\) = \(-g^{\mu \rho} g_{\rho \sigma} \nabla_{\alpha} g^{\nu \sigma}\) = \(-\nabla_{\alpha} g^{\mu \nu}\)
\(Q\) = – \( Q_{\alpha \mu \nu}P^{\alpha \mu \nu}\)
\(Q = – \frac{1}{4}[\nabla_\alpha g_{\mu \nu} \nabla^\alpha g^{\mu \nu} – 2 \nabla_\alpha g_{\mu \nu} \nabla^\mu g^{\alpha \nu} + (g_{\rho \mu} \nabla_\alpha g^{\rho \mu})(g_{\sigma \nu} \nabla^\alpha g^{\sigma \nu}) – 2(g_{\mu \rho} \nabla_\alpha g^{\mu \rho})(\nabla_\beta g^{\alpha \beta})]\)
where \(Q_\alpha = -g_{\rho \mu} \nabla_\alpha g^{\rho \mu}\), \(Q^\alpha = -g_{\sigma \nu} \nabla^{\alpha} g^{\sigma \nu}\), \(\tilde{Q}_{\alpha} = \nabla^{\beta} g_{\alpha \beta}\) and \(\tilde{Q}^{\alpha} = \nabla_{\beta} g^{\alpha \beta}\)
\(P^{\alpha \mu}_{\ \ \nu} = \) Find Yourself
\(P^\alpha_{\mu \nu} = \frac{1}{4}[-Q^\alpha_{\mu \nu} + 2 Q_{(\mu \ \ \ \ \nu)}^{\ \ \ \ \ \alpha} +Q^\alpha g_{\mu \nu} \, \, – \, \, \tilde{Q^\alpha}g_{\mu \nu} \, \, – \, \, \frac{1}{2}(\delta^\alpha_\mu Q_\nu + \delta^\alpha_\nu Q_\mu)]\)
\(P^{\alpha \mu \nu} = \frac{1}{4}[-Q^{\alpha \mu \nu} + Q^{\mu \alpha \nu} + Q^{\nu \alpha \mu} + Q^\alpha g^{\mu \nu} – \tilde{Q^\alpha} g^{\mu \nu} – \frac{1}{2}(g^{\alpha \mu}Q^\nu + g^{\alpha \nu}Q^\mu)]\)
\(P_{\alpha \nu \rho} = – \frac{1}{4}[Q_{\alpha \nu \rho} \, \, – \, \, Q_{\rho \alpha \nu} \, \, – \, \, Q_{\nu \rho \alpha} + (\tilde{Q_\alpha} \, \, – \, \, Q_\alpha)g_{\nu \rho} \, + \, Q_\nu g_{\alpha \rho}]\)
