BAO constraints on dark energy parameters

The BAO in the radial and transverse directions provide measurements of the Hubble parameter and angular diameter distance, respectively. The angular diameter distance and Hubble parameter can include different functions that explain dark energy behavior.^[23][24] These functions have two parameters w₀ and w₁ and one can constrain them with a chi-square technique.^[25]

General relativity and dark energy

In general relativity, the expansion of the universe is parametrized by a scale factor ${\displaystyle a(t)}$ which is related to redshift:^[4]

${\displaystyle a(t)\equiv (1+z(t))^{-1}\!}$

The Hubble parameter, ${\displaystyle H(z)}$, in terms of the scale factor is:

${\displaystyle H(t)\equiv {\frac {\dot {a}}{a}}\!}$

where ${\displaystyle {\dot {a}}}$ is the time-derivative of the scale factor. The Friedmann equations express the expansion of the universe in terms of Newton's gravitational constant, ${\displaystyle G}$, the mean gauge pressure, ${\displaystyle p}$, the Universe's density ${\displaystyle \rho \!}$, the curvature, ${\displaystyle k}$, and the cosmological constant, ${\displaystyle \Lambda \!}$:^[4]

${\displaystyle H^{2}=\left({\frac {\dot {a}}{a}}\right)^{2}={\frac {8\pi G}{3}}\rho -{\frac {kc^{2}}{a^{2}}}+{\frac {\Lambda c^{2}}{3}}}$

${\displaystyle {\dot {H}}+H^{2}={\frac {\ddot {a}}{a}}=-{\frac {4\pi G}{3}}\left(\rho +{\frac {3p}{c^{2}}}\right)+{\frac {\Lambda c^{2}}{3}}}$

Observational evidence of the acceleration of the universe implies that (at present time) ${\displaystyle {\ddot {a}}>0}$. Therefore, the following are possible explanations:^[26]

• The universe is dominated by some field or particle that has negative pressure such that the equation of state:

${\displaystyle w={\frac {p}{\rho }}<-1/3\!}$

• There is a non-zero cosmological constant, ${\displaystyle \Lambda \!}$.
• The Friedmann equations are incorrect since they contain oversimplifications in order to make the general relativistic field equations easier to compute.

In order to differentiate between these scenarios, precise measurements of the Hubble parameter as a function of redshift are needed.

Measured observables of dark energy

The density parameter, ${\displaystyle \Omega \!}$, of various components, ${\displaystyle x}$, of the universe can be expressed as ratios of the density of ${\displaystyle x}$ to the critical density, ${\displaystyle \rho _{c}\!}$:^[26]

${\displaystyle \rho _{c}={\frac {3H^{2}}{8\pi G}}}$

${\displaystyle \Omega _{x}\equiv {\frac {\rho _{x}}{\rho _{c}}}={\frac {8\pi G\rho _{x}}{3H^{2}}}}$

The Friedman equation can be rewritten in terms of the density parameter. For the current prevailing model of the universe, ΛCDM, this equation is as follows:^[26]

${\displaystyle H^{2}(a)=\left({\frac {\dot {a}}{a}}\right)^{2}=H_{0}^{2}\left[\Omega _{m}a^{-3}+\Omega _{r}a^{-4}+\Omega _{k}a^{-2}+\Omega _{\Lambda }a^{-3(1+w)}\right]}$

where m is matter, r is radiation, k is curvature, Λ is dark energy, and w is the equation of state. Measurements of the CMB from WMAP put tight constraints on many of these parameters; however it is important to confirm and further constrain them using an independent method with different systematics.

The BAO signal is a standard ruler such that the length of the sound horizon can be measured as a function of cosmic time.^[3] This measures two cosmological distances: the Hubble parameter, ${\displaystyle H(z)}$, and the angular diameter distance, ${\displaystyle d_{A}(z)}$, as a function of redshift ${\displaystyle (z)}$.^[27] By measuring the subtended angle, ${\displaystyle \Delta \theta }$, of the ruler of length ${\displaystyle \Delta \chi }$, these parameters are determined as follows:^[27]

${\displaystyle \Delta \theta ={\frac {\Delta \chi }{d_{A}(z)}}\!}$

${\displaystyle d_{A}(z)\propto \int _{0}^{z}{\frac {dz'}{H(z')}}\!}$

the redshift interval, ${\displaystyle \Delta z}$, can be measured from the data and thus determining the Hubble parameter as a function of redshift:

${\displaystyle c\Delta z=H(z)\Delta \chi \!}$

Therefore, the BAO technique helps constrain cosmological parameters and provide further insight into the nature of dark energy.