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====== Baryon Acoustic Oscillation ======



Baryon Acoustic Oscillations (BAO) refer to the regular, periodic fluctuations in the density of the visible baryonic matter of the Universe. Much like how sound waves travel through the air, these acoustic waves traveled through the primordial plasma of the early universe. Today, BAO provides a robust "standard ruler" for cosmology, allowing us to measure the expansion history of the Universe and constrain the properties of Dark Energy.

===== The Physics of the Early Universe =====

Prior to recombination (at redshift $z \approx 1100$), the Universe was a hot, dense plasma consisting of photons, baryons (protons and electrons), and dark matter. 

**Dark Matter:** Cold and collisionless. It responds only to gravity, initiating the collapse of primordial overdensities seeded by quantum fluctuations during inflation.\\
**Photon-Baryon Fluid:** Electrons and photons were tightly coupled via Thomson scattering, and protons were coupled to electrons via Coulomb interactions. This created a single, relativistic photon-baryon fluid.

When a dark matter overdensity collapsed under gravity, it pulled the photon-baryon fluid with it. However, the immense radiation pressure of the photons strongly resisted this compression, driving the fluid outward. **This competition between gravity and radiation pressure generated spherical acoustic waves propagating away from the initial overdensities.**




===== Mathematical Formalism of the Acoustic Waves =====

The dynamics of the photon-baryon fluid can be described by cosmological perturbation theory. Ignoring metric perturbations (the Sachs-Wolfe effect) for simplicity, the evolution of the temperature/density perturbations is governed by the driven wave equation.

**The Sound Speed:**\\
The speed at which these acoustic waves propagate is the sound speed of the photon-baryon fluid, $c_s$. It is determined by the ratio of the fluid's pressure $p$ to its energy density $\rho$:

$$c_s^2 = \frac{\delta p}{\delta \rho} = \frac{\delta p_\gamma + \delta p_b}{\delta \rho_\gamma + \delta \rho_b}$$

Since baryons are non-relativistic, their pressure is negligible ($p_b \approx 0$). The photon pressure is $p_\gamma = \rho_\gamma c^2 / 3$. Defining the baryon-to-photon momentum density ratio $R$:

$$R \equiv \frac{3\rho_b}{4\rho_\gamma}$$

The sound speed evaluates to:

$$c_s = \frac{c}{\sqrt{3(1+R)}}$$

In the very early Universe, radiation dominates ($R \to 0$), and the sound speed approaches the relativistic limit $c/\sqrt{3}$. As the Universe expands and baryons begin to contribute more to the inertia of the fluid, $c_s$ drops.



[{{ :courses:ast403:bao_wave.gif | Fig 1: A gif animation showing the effect two opposing effect.}}]


**The Comoving Sound Horizon:
**\\
The most critical scale in BAO cosmology is the comoving sound horizon, $r_s$. This is the total comoving distance a sound wave could have traveled from the Big Bang ($z = \infty$) until the drag epoch ($z_d \approx 1059$), when photons completely decoupled from baryons, and the wave stalled.

The comoving sound horizon is given by the integral of the sound speed over conformal time $d\eta = dt/a$:

$$r_s = \int_0^{\eta_d} c_s(\eta) d\eta = \int_{z_d}^{\infty} \frac{c_s(z)}{H(z)} dz$$

Where $H(z)$ is the Hubble parameter at redshift $z$. Current observations from the Cosmic Microwave Background (CMB) place $r_s$ at approximately $147$ Mpc (megaparsecs). 

===== Recombination and the Relic Signature =====

At recombination, the temperature of the Universe dropped sufficiently for electrons and protons to form neutral hydrogen. Thomson scattering ceased.\\ 
The **photons** streamed away freely, carrying the imprint of these temperature fluctuations to us as the Cosmic Microwave Background (CMB).\\
The **baryons**, having lost the radiation pressure driving them outward, stalled. They formed a spherical shell of slight overdensity at exactly the radius of the sound horizon ($r_s \approx 147$ Mpc) surrounding the central dark matter halo.\\

Over billions of years, both the central dark matter peak and the spherical baryonic shell acted as gravitational seeds for structure formation. Therefore, for any given galaxy, there is a slightly enhanced probability of finding another galaxy exactly $147$ Mpc away.

[{{ :courses:ast403:bao.png?600 | Fig 2: Artist's impression of BAO signature. Credit: ESA and the Planck Collaboration / Gabriela Secara / Perimeter Institute}}]


===== Observational Signatures: The Standard Ruler =====

Because the scale of the sound horizon $r_s$ is firmly rooted in the well-understood physics of the early Universe, it serves as a cosmological Standard Ruler. By observing the apparent size of the BAO scale in the clustering of galaxies at different redshifts, we can measure the geometry and expansion rate of the Universe.

We measure the BAO signal in two directions relative to our line of sight:

**Transverse Direction:** Measures the angular diameter distance $D_A(z)$. If the BAO feature subtends an angle $\Delta \theta$ on the sky, then:

$$\Delta \theta = \frac{r_s}{(1+z)D_A(z)}$$
    
**Line-of-Sight Direction:** Measures the Hubble parameter $H(z)$ directly. The BAO scale appears as a redshift interval $\Delta z$:
$$\Delta z = \frac{H(z) r_s}{c}$$

By mapping the 3D distribution of millions of galaxies (via surveys like SDSS, DESI, and Euclid) and measuring the 2-point correlation function $\xi(r)$ or the matter power spectrum $P(k)$, cosmologists can isolate the BAO peak. Measuring $D_A(z)$ and $H(z)$ across cosmic time tightly constrains the equation of state of dark energy, $w$.


[{{ :courses:ast403:desi_bao.jpg?600 | Fig 3: Large-scale structure from DESI redshift survey}}]


An animation of the formation of BAO: [[https://www.youtube.com/watch?v=jpXuYc-wzk4]]