Merge branch 'JB_changes_cosmology_theory' into 'cosmology'

Minor proofreading changes

See merge request !496

theory/Cosmology/coordinates.tex

 \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],

theory/Cosmology/flrw.tex

 \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 $\gamma = 5/3$.