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Equation of State and Gravitational Wave

Update: The equation of state parameter \( w \) data was newly added on 2024-07-19.

Tabulated data of the effective degrees of freedom and the transfer function estimated in this paper are distributed.

You can get the data for the effective degrees of freedom \( g_{*\rho,s} \) here and the equation of state parameter \( w \) here.

We also provide a C++ code for the fitting functions.

The data for the transfer function are available here.

If you find any problems or request additional data, please contact me (satoshi.shirai_at_ipmu.jp).


Equation of State

The effective degrees of freedom are defined as: \[ g_{*\rho}(T) \equiv \frac{\rho(T)}{\left[\frac{\pi^2 T^4}{30}\right]},\quad g_{*s}(T) \equiv \frac{s(T)}{\left[\frac{2\pi^2 T^3}{45}\right]}. \] The equation of state parameter is given by \[ w(T) \equiv \frac{p(T)}{\rho(T)}. \]
Effective degrees of freedom

Effective degrees of freedom.
EOS

Equation of state.

Transfer function

The tranfer function is defined as: \[ T(k) \equiv \frac{1}{12a_0^2 H_0^2}\overline{\left[\chi'(\tau_0,k)\right]^2}h^2. \] Using this function, you can produce the spectrum of primordial GWs for a given primordial tensor power spectrum \(\mathcal{P}_T(k)\). \[ \Omega_{\rm gw}h^2(k) = T(k)\mathcal{P}_T(k). \]
Transfer function

Transfer function.