I want to learn about cosmology, maybe start with the book “An Introduction to Modern Cosmology” by Andrew Liddle and the lecture “The Early Universe” by MIT OpenCourseWare.
The standard Big Bang
It does not tell the cause of the Bang, but the aftermath of the Bang. It assumes all the matter already existed before the Bang.
Cosmic Inflation
A prequal to the Big Bang.
- gravity can be repulsive when pressure is negative. (Find out more about the relationship between gravity and pressure in the next part.)
There is a patch of repulsive gravity material in the early universe, which is the cause of the Big Bang.
The (mass/energy) density of the repulsive material is not lowered as it expands.The resolution is to introduce negative energy.
Complementarity: The relationship between gravity and pressure
Below is a high-level introduction to these ideas, followed by some of the key formulas in cosmology and general relativity that show why negative pressure can produce a repulsive gravitational effect.
1. Stress-Energy Tensor and Einstein’s Field Equations
In general relativity (GR), gravity is described by Einstein’s field equations, which relate the geometry of spacetime (through the Einstein tensor \(G_{\mu\nu}\)) to the energy and momentum content of spacetime (through the stress-energy tensor \(T_{\mu\nu}\)):
\[ G_{\mu\nu} \;=\; \frac{8\pi G}{c^4}\,T_{\mu\nu}, \]where \(G\) is the gravitational constant, and \(c\) is the speed of light. The left-hand side encapsulates the curvature of spacetime, while the right-hand side tells us how mass, energy, momentum, and pressure determine that curvature.
Perfect Fluid Approximation
In cosmology, we often model the contents of the universe (whether it be matter, radiation, or an inflaton field) as a “perfect fluid,” whose stress-energy tensor is written as:
\[ T_{\mu\nu} \;=\; (\rho + p)\,u_{\mu}\,u_{\nu} \;+\; p\,g_{\mu\nu}. \]- \(\rho\) is the energy density.
- \(p\) is the pressure.
- \(u_\mu\) is the 4-velocity of the fluid.
- \(g_{\mu\nu}\) is the metric tensor of spacetime.
This formula captures the idea that both \(\rho\) and \(p\) play significant roles in determining spacetime curvature, not just the mass or energy density alone.
2. Friedmann–Lemaître–Robertson–Walker (FLRW) Cosmology
For cosmological applications, we often assume a homogeneous, isotropic universe described by the FLRW metric. Two key equations (the Friedmann equations) emerge from Einstein’s field equations under this assumption:
Friedmann Equation (for the expansion rate \(H = \dot{a}/a\)):
\[ H^2 \;=\; \left(\frac{\dot{a}}{a}\right)^2 \;=\; \frac{8\pi G}{3}\,\rho \;-\; \frac{k}{a^2} \;+\; \frac{\Lambda}{3}, \]where
- \(a(t)\) is the scale factor describing how distances in the universe scale with time,
- \(k\) is a parameter related to spatial curvature,
- \(\Lambda\) is the cosmological constant (sometimes included on the right side instead).
Acceleration Equation (for the second derivative \(\ddot{a}\)):
\[ \frac{\ddot{a}}{a} \;=\; -\frac{4\pi G}{3}\,(\rho + 3p) \;+\; \frac{\Lambda}{3}. \]This equation is especially telling: the combination \(\rho + 3p\) directly governs whether the universe’s expansion accelerates or decelerates.
3. Repulsive Gravity and Negative Pressure
Notice in the acceleration equation:
\[ \frac{\ddot{a}}{a} \;=\; -\frac{4\pi G}{3}(\rho + 3p) + \frac{\Lambda}{3}. \]- If \(\rho + 3p > 0\), the term \(-\frac{4\pi G}{3}(\rho + 3p)\) is negative, slowing the expansion.
- If \(\rho + 3p < 0\), this term becomes positive, contributing to \(\ddot{a} > 0\), i.e. accelerated expansion—often described as a repulsive gravitational effect.
In simpler terms, because pressure enters the equations with a factor of 3 compared to density, a sufficiently large negative pressure (\(p < 0\)) can overpower \(\rho\) in \(\rho + 3p\) and make the combination negative. This is precisely what happens during cosmic inflation (driven by the “inflaton field” with an effective negative pressure) and in scenarios with a cosmological constant \(\Lambda\) (dark energy) where the vacuum energy acts like negative pressure.
4. The Inflaton Field and Negative Pressure
During inflation, the universe is dominated by an inflaton field whose energy density \(\rho_{\phi}\) remains nearly constant, while its effective pressure \(p_{\phi}\) is negative. (In fact, for a slowly rolling scalar field, \(p_{\phi} \approx -\rho_{\phi}\) at times.) This makes:
\[ \rho_{\phi} + 3p_{\phi} \; \approx \; \rho_{\phi} + 3(-\rho_{\phi}) \;=\; \rho_{\phi} - 3\rho_{\phi} \;=\; -2\rho_{\phi} < 0, \]leading to \(\ddot{a} > 0\), and thus an exponentially accelerating expansion (repulsive effect).
Putting It All Together
- Einstein’s field equations show that pressure as well as energy density affect spacetime curvature.
- Negative pressure can cause a runaway expansion (i.e., the scale factor \(a(t)\) grows at an accelerating rate).
- Inflation is an example of this effect, where the inflaton’s negative pressure dominates and drives exponential growth of the universe.
In summary:
- Gravity is not just about mass. Energy density \(\rho\) and pressure \(p\) both appear in the stress-energy tensor.
- Repulsive gravity arises when \(\rho + 3p < 0\). This condition makes the cosmic scale factor’s acceleration positive (\(\ddot{a} > 0\)), indicating accelerated expansion.
Additional Thoughts
Where does the negative pressure come from? There’s not a determined theory yet. Some suggests there should be some particles called inflatons. From a classical point of view, it’s a senario that when a particle flies into a wall, it doesn’t bounce back, but it goes through the wall and comes out from the other side. This is a quantum effect. The inflaton field is a quantum field that has a negative pressure.
In next post, I will review the key concepts of Einstein’s field equations and the meaning of those tensors.
Last modified on 2024-12-23