BAO_forecast

Forecasting Baryon Acoustic Oscillation (BAO) distance measurement precision for photometric galaxy surveys using Fisher matrix methods based on Seo & Eisenstein (2007).

Seo & Eisenstein (2007) Improved Forecasts for the Baryon Acoustic Oscillations and Cosmological Distance Scale https://arxiv.org/abs/astro-ph/0701079

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Features

• BAO Fisher forecasting in individual redshift bins and combined precision.
• Automatic cosmology handling via cosmoprimo.
• Includes:
  • redshift-space distortions (RSD),
  • Silk damping,
  • non-linear BAO damping,
  • photo-z smearing.
• Returns:
  • transverse distance precision: $$\sigma_D\equiv\sigma(\ln D_M).$$

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Inputs

The main input to the forecast consists of:

• redshift-bin edges $(z_{\min}, z_{\max})$.
• galaxy number in each bin $N_{\rm gal}$.
• photometric redshift uncertainty $\sigma_z$.
• survey area $A$ (in square degrees).

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Forecast Formalism

The BAO Fisher matrix is computed as a double integral over wavenumber $k$ and angle cosine $\mu$:

$$F_{ij} = \frac{A_{\rm BAO}^2 V}{2} \int d\mu \int dk \frac{k^2\ f_i(\mu)\ f_j(\mu)\ \exp[-2(k\Sigma_{\rm Silk})^{1.4}]\ \exp[-k^2 \Sigma_{\rm nl}^2(\mu)]}{\left[P_{\rm BAO}(k) + \frac{1}{nP\ R(k,\mu)}\right]^2},$$

where

• the angular response functions are

$$f_D(\mu)=1-\mu^2,\quad f_H(\mu)=\mu^2.$$

• the volume is given by

$$V = \frac{4\pi}{3} f_{\rm sky} \left(\chi_{\rm max}^3 - \chi_{\rm min}^3\right),\quad\text{where}\quad f_{\rm sky} = \frac{A(\text{deg}^2)}{4\pi (180/\pi)^2}.$$

• the shotnoise parameter $nP$ is given by

$$nP = \bar{n} P_{\rm eff},\quad\text{where}\quad\bar{n} = \frac{N_{\rm gal}}{V}\quad\text{and}\quad P_{\rm eff} = (bD)^2 P(k=0.2\ \text{Mpc}/h).$$

• the redshift-space distortion factor is

$$R(k,\mu)=\left(1+\beta\mu^2\right)^2 \exp[-k^2\mu^2\Sigma_z^2],\quad\text{where}\quad\Sigma_z = \frac{c}{H(z)}\sigma_z.$$

• the anisotropic nonlinear damping is

$$\Sigma_{\rm nl}^2(\mu)=\Sigma_\parallel^2\mu^2+\Sigma_\perp^2(1-\mu^2),\quad \text{where}\quad \Sigma_\parallel=\Sigma_0 D(1+f)\quad\text{and}\quad\Sigma_\perp=\Sigma_0 D$$

The correlation coefficient between radial and transverse modes is, then, computed as

$$r=\frac{F_{DH}}{\sqrt{F_{DD}F_{HH}}},$$

and the angular BAO distance uncertainty is

$$\sigma_D=\frac{1}{\sqrt{F_{DD}(1-r^2)}}.$$

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Requirements

• numpy
• cosmoprimo