diff --git a/theory/Cosmology/coordinates.tex b/theory/Cosmology/coordinates.tex index 4cf455e367608d93e10bc654f2ff392159571770..83deb5dc59a3af48e016516f58971fc93971a23b 100644 --- a/theory/Cosmology/coordinates.tex +++ b/theory/Cosmology/coordinates.tex @@ -1,10 +1,10 @@ \subsection{Choice of co-moving coordinates} \label{ssec:ccordinates} -Note that unlike the \gadget convention we do not express quantities -with ``little h'' included. We hence use $\rm{Mpc}$ and not -${\rm{Mpc}}/h$ for instance. Similarly, the time integration operators -(see below) also include an $h$-factor via the explicit appearance of +Note that, unlike the \gadget convention, we do not express quantities with +``little h'' ($h$) included; for instance units of length are expressed in +units of $\rm{Mpc}$ and not ${\rm{Mpc}}/h$. Similarly, the time integration +operators (see below) also include an $h$-factor via the explicit appearance of the Hubble constant.\\ In physical coordinates, the Lagrangian for a particle $i$ in the \cite{Springel2002} flavour of SPH with gravity reads @@ -15,7 +15,7 @@ In physical coordinates, the Lagrangian for a particle $i$ in the m_i \phi \end{equation} Introducing the comoving positions $\mathbf{r}'$ such that $\mathbf{r} -= a(t) \mathbf{r}'$ and comoving densities $\rho' \equiv a^3(t)\rho$ ,we get += a(t) \mathbf{r}'$ and comoving densities $\rho' \equiv a^3(t)\rho$, \begin{equation} \Lag = \frac{1}{2} m_i \left(a\dot{\mathbf{r}}_i' + \dot{a}\mathbf{r}_i' @@ -23,8 +23,8 @@ Introducing the comoving positions $\mathbf{r}'$ such that $\mathbf{r} \frac{1}{\gamma-1}m_iA_i'\left(\frac{\rho_i'}{a^3}\right)^{\gamma-1} - m_i \phi, \end{equation} -where we chose to define $A'=A$ such that the equation of state for -the gas and thermodynamics relations between quantities have the same +where $A'=A$ is chosen such that the equation of state for +the gas and thermodynamic relations between quantities have the same form (i.e. are scale-factor free) in the primed coordinates as well. This implies \begin{equation} @@ -41,22 +41,23 @@ $\Psi \equiv \frac{1}{2}a\dot{a}\mathbf{r}_i^2$ and obtain \phi' &= a\phi + \frac{1}{2}a^2\ddot{a}\mathbf{r}_i'^2.\nonumber \end{align} Finally, we introduce the velocities $\mathbf{v}' \equiv -a^2\dot{\mathbf{r}'}$ used internally by the code. Note that these -velocities do not have a physical interpretation. We caution that they +a^2\dot{\mathbf{r}'}$ that are used internally by the code. Note that these +velocities \emph{do not} have a physical interpretation. We caution that they are not the peculiar velocities, nor the Hubble flow, nor the total velocities\footnote{One additional inconvenience of our choice of generalised coordinates is that our velocities $\mathbf{v}'$ and sound-speed $c'$ do not have the same dependencies on the scale-factor. The signal velocity entering the time-step calculation will hence read $v_{\rm sig} = a\dot{\mathbf{r}'} + c = - \frac{|\mathbf{v}'|}{a} + a^{1-3(\gamma-1)/2}c'$.}. Using the SPH + \frac{|\mathbf{v}'|}{a} + a^{1-3(\gamma-1)/2}c'$.}. +This choice implies that $\dot{v}' = a \ddot{r}$. Using the SPH definition of density, $\rho_i = \sum_jm_jW(\mathbf{r}_{j}'-\mathbf{r}_{i}',h_i') = \sum_jm_jW_{ij}'(h_i')$, we can follow \cite{Price2012} and apply the Euler-Lagrange equations to write \begin{alignat}{3} \dot{\mathbf{r}}_i'&= \frac{1}{a^2} \mathbf{v}_i'& \label{eq:cosmo_eom_r} \\ - \dot{\mathbf{v}}_i' &= \sum_j m_j &&\left[\frac{1}{a^{3(\gamma-1)}}f_i'A_i'\rho_i'^{\gamma-2}\mathbf{\nabla}_i'W_{ij}'(h_i)\right. \nonumber\\ + \dot{\mathbf{v}}_i' &= -\sum_j m_j &&\left[\frac{1}{a^{3(\gamma-1)}}f_i'A_i'\rho_i'^{\gamma-2}\mathbf{\nabla}_i'W_{ij}'(h_i)\right. \nonumber\\ & && + \left. \frac{1}{a^{3(\gamma-1)}}f_j'A_j'\rho_j'^{\gamma-2}\mathbf{\nabla}_i'W_{ij}'(h_j)\right. \nonumber\\ & && + \left. \frac{1}{a}\mathbf{\nabla}_i'\phi'\right] \label{eq:cosmo_eom_v} \end{alignat} @@ -66,11 +67,11 @@ with \rho_i'}{\partial h_i'}\right]^{-1}, \qquad \mathbf{\nabla}_i' \equiv \frac{\partial}{\partial \mathbf{r}_{i}'}. \nonumber \end{equation} -These corresponds to the equations of motion for density-entropy SPH +These correspond to the equations of motion for density-entropy SPH \citep[e.g. eq. 14 of][]{Hopkins2013} with cosmological and gravitational terms. SPH flavours that evolve the internal energy $u$ instead of the -entropy the additional equation of motion describing the evolution of -$u'$ becomes: +entropy require the additional equation of motion describing the evolution of +$u'$: \begin{equation} \dot{u}_i' = \frac{P_i'}{\rho_i'^2}\left[3H\rho_i' + \frac{1}{a^2}f_i'\sum_jm_j\left(\mathbf{v}_i' - \mathbf{v}_j'\right)\cdot\mathbf{\nabla}_i'W_{ij}'(h_i)\right], diff --git a/theory/Cosmology/flrw.tex b/theory/Cosmology/flrw.tex index af504159420181ca83d545892e43b054ba83be4a..8b429d14acb685317c5394c60c303321c10a33bb 100644 --- a/theory/Cosmology/flrw.tex +++ b/theory/Cosmology/flrw.tex @@ -1,58 +1,54 @@ \subsection{Background evolution} \label{ssec:flrw} -In \swift we assume a standard FLRW metric for the evolution of the -background density of the Universe and use the Friedmann equations to -describe the evolution of the scale-factor $a(t)$. As always, we -scale $a$ such that its value at present-day is $a_0 \equiv a(t=t_{\rm - now}) = 1$. We also define redshift $z \equiv 1/a - 1$ and the -Hubble parameter at a given redshift +In \swift we assume a standard FLRW metric for the evolution of the background +density of the Universe and use the Friedmann equations to describe the +evolution of the scale-factor $a(t)$. We scale $a$ such that its present-day +value is $a_0 \equiv a(t=t_{\rm now}) = 1$. We also define redshift $z \equiv +1/a - 1$ and the Hubble parameter \begin{equation} H(t) \equiv \frac{\dot{a}(t)}{a(t)} \end{equation} -with its present-day value denoted as $H_0 = H(t=t_{\rm - now})$. Following normal conventions we write $H_0 = 100 -h~\rm{km}\cdot\rm{s}^{-1}\cdot\rm{Mpc}^{-1}$ and use $h$ as the input -parameter for the Hubble constant. +with its present-day value denoted as $H_0 = H(t=t_{\rm now})$. Following +normal conventions we write $H_0 = 100 +h~\rm{km}\cdot\rm{s}^{-1}\cdot\rm{Mpc}^{-1}$ and use $h$ as the input parameter +for the Hubble constant. -To allow for general expansion histories we use the full Friedmann -equations and write +To allow for general expansion histories we use the full Friedmann equations +and write \begin{align} H(a) &\equiv H_0 E(a) \\ E(a) &\equiv\sqrt{\Omega_m a^{-3} + \Omega_r a^{-4} + \Omega_k a^{-2} + \Omega_\Lambda a^{-3(1+w(a))}}, \label{eq:friedmann} \end{align} -where the dark energy equation-of-state is evolving according to the -formulation of \cite{Linder2003}: +where the dark energy equation-of-state evolves according to the formulation of +\cite{Linder2003}: \begin{equation} w(a) \equiv w_0 + w_a~(1-a). \end{equation} -The cosmological model is hence fully defined by specifying the -dimensionless constants $\Omega_m$, $\Omega_r$, $\Omega_k$, -$\Omega_\Lambda$, $h$, $w_0$ and $w_a$ as well as the starting -redshift (or scale-factor of the simulation) $a_{\rm start}$ and final -time $a_{\rm end}$. \\ At any scale-factor $a_{\rm age}$, the time -$t_{\rm age}$ since the Big Bang (age of the Universe) can be computed -as \citep[e.g.][]{Wright2006}: +The cosmological model is hence fully defined by specifying the dimensionless +constants $\Omega_m$, $\Omega_r$, $\Omega_k$, $\Omega_\Lambda$, $h$, $w_0$ and +$w_a$ as well as the starting redshift (or scale-factor of the simulation) +$a_{\rm start}$ and final time $a_{\rm end}$. \\ At any scale-factor $a_{\rm +age}$, the time $t_{\rm age}$ since the Big Bang (age of the Universe) can be +computed as \citep[e.g.][]{Wright2006}: \begin{equation} t_{\rm age} = \int_{0}^{a_{\rm age}} dt = \int_{0}^{a_{\rm age}} \frac{da}{a H(a)} = \frac{1}{H_0} \int_{0}^{a_{\rm age}} \frac{da}{a E(a)}. \end{equation} -For a general set of cosmological parameters, this integral can only -be evaluated numerically, which is too slow to be evaluated accurately -during a run. At the start of the simulation we, hence, evaluate this -integral for $10^4$ values of $a_{\rm age}$ equally spaced between -$\log(a_{\rm start})$ and $\log(a_{\rm end})$. These are obtained via -adaptive quadrature using the 61-points Gauss-Konrod rule implemented -in the {\sc gsl} library \citep{GSL} with a relative error limit of -$\epsilon=10^{-10}$. The value for a specific $a$ (over the course of -a simulation run) is then obtained by linear interpolation of that +For a general set of cosmological parameters, this integral can only be +evaluated numerically, which is too slow to be evaluated accurately during a +run. At the start of the simulation we tabulate this integral for $10^4$ values +of $a_{\rm age}$ equally spaced between $\log(a_{\rm start})$ and $\log(a_{\rm +end})$. The values are obtained via adaptive quadrature using the 61-points +Gauss-Konrod rule implemented in the {\sc gsl} library \citep{GSL} with a +relative error limit of $\epsilon=10^{-10}$. The value for a specific $a$ (over +the course of a simulation run) is then obtained by linear interpolation of the table. -\subsubsection{NOTES FOR PEDRO} +\subsubsection{Typical Values of the Cosmological Parameters} -Typical values for the constants are: $\Omega_m = 0.3, -\Omega_\Lambda=0.7, 0 < \Omega_r<10^{-3}, |\Omega_k | < 10^{-2}, -h=0.7, a_{\rm start} = 10^{-2}, a_{\rm end} = 1, w_0 = -1\pm 0.1, -w_a=0\pm0.2$ and $\gamma = 5/3$. +Typical values for the constants are: $\Omega_m = 0.3, \Omega_\Lambda=0.7, 0 < +\Omega_r<10^{-3}, |\Omega_k | < 10^{-2}, h=0.7, a_{\rm start} = 10^{-2}, a_{\rm +end} = 1, w_0 = -1\pm 0.1, w_a=0\pm0.2$ and $\gamma = 5/3$.