Fix various typos and other small mistakes, and use ~ for the references and citations

Author: AgenttiX

create_arxiv.sh

@@ -24,7 +24,8 @@ mkdir arxiv/anc
 # Rsync is used to enable excluding.
 rsync -av --exclude='fig/*-converted-to.pdf' --exclude='fig/lecture_notes' --exclude='fig/tampere' --exclude='tex/other_models.tex' \
   fig tex \
-  .latexmkrc babelbst.tex biblatex-dm.cfg englbst.tex finnbst.tex LICENSE main.bbl main.tex Makefile swedbst.tex tktl.bst UH-logo.png UH_TCM_MSc.cls \
+  .latexmkrc babelbst.tex biblatex-dm.cfg englbst.tex finnbst.tex LICENSE \
+  main.bbl main.tex Makefile swedbst.tex tktl.bst UH-logo.png UH_TCM_MSc.cls \
   arxiv
 # Ancillary files are placed in the anc directory.
 # https://info.arxiv.org/help/ancillary_files.html

main.tex

@@ -94,7 +94,7 @@
 % - Project readme
 % The \today macro should not be used on arXiv.
 % https://info.arxiv.org/help/faq/today.html
-\date{18th December 2024}
+\date{18th December 2024, updated on 9th June 2025}
 \prof{Professor Mark Hindmarsh}
 \censors{Professor Mark Hindmarsh}{Professor Kari Rummukainen}{}
 \depositeplace{arXiv, \href{https://hdl.handle.net/10138/591514}{Helda}, \href{https://github.com/AgenttiX/msc-thesis2}{GitHub}}

tex/bag.tex

@@ -11,42 +11,45 @@ \section{The bag model}
 This results in the bag equation of state
 \cites[eq. 7.33]{lecture_notes}[eq. 8-9]{giese_2020}
 \begin{equation}
-p_\pm = a_\pm T^4 - V_\pm,
+p_\pm = a_\pm T^4 - V_\pm.
 \label{eq:bag_p}
 \end{equation}
-The potential difference $\Delta V \equiv V_+ - V_-$, also known as $\epsilon$, is the temperature-independent vacuum energy that is released in the phase transition.
+The potential difference $\Delta V \equiv V_+ - V_-$, also known as $\epsilon$,
+is the temperature-independent vacuum energy that is released in the phase transition.
 The potentials are usually shifted so that $V_b = 0$.
-The name of the bag model originates from quantum chromodynamics (QCD), where it's used to describe the proton as a bag of quarks.
+The name of the bag model originates from quantum chromodynamics (QCD),
+where it's used to describe the proton as a bag of quarks.
 \cite{giese_2020}
 
-Using eq. \eqref{eq:entropy_density} we get the entropy density
+Using eq.~\eqref{eq:entropy_density} we get the entropy density
 \begin{equation}
 s_\pm = 4 a_\pm T^3,
 \end{equation}
-and with eq. \eqref{eq:enthalpy_entropy} the enthalpy density
+and with eq.~\eqref{eq:enthalpy_entropy} the enthalpy density
 \begin{equation}
 w_\pm = 4 a_\pm T^4.
 \end{equation}
-Finally with eq. \eqref{eq:enthalpy_sum} we get the energy density
+Finally with eq.~\eqref{eq:enthalpy_sum} we get the energy density
 \begin{equation}
-e_\pm = 3 a_\pm T^4 + V_\pm
+e_\pm = 3 a_\pm T^4 + V_\pm.
 \end{equation}
-The speed of sound from eq. \eqref{eq:cs2_explicit} simplifies to
+The speed of sound from eq.~\eqref{eq:cs2_explicit} simplifies to
 \begin{equation}
 c_s^2 = \frac{dp}{de} = \frac{dp/dT}{de/dT} = \frac{1}{3}.
 \end{equation}
-As in section \ref{general_eq}, this corresponds to assuming both of the phases to be ultrarelativistic.
-It also happens to be the same as the $\tilde{v}_-$ that corresponds to eq. \eqref{eq:v_tilde_plus_limit}, and therefore the Chapman-Jouguet speed $v_{CJ}$ of the bag model is given by eq. \eqref{eq:v_tilde_plus_limit}.
-The trace anomaly of eq. \eqref{eq:theta} simplifies to
+As in section~\ref{general_eq}, this corresponds to assuming both of the phases to be ultrarelativistic.
+It also happens to be the same as the $\tilde{v}_-$ that corresponds to eq.~\eqref{eq:v_tilde_plus_limit},
+and therefore the Chapman-Jouguet speed $v_{CJ}$ of the bag model is given by eq.~\eqref{eq:v_tilde_plus_limit}.
+The trace anomaly of eq.~\eqref{eq:theta} simplifies to
 \begin{align}
 \theta_\pm = V_\pm.
 \end{align}
-Therefore the transition strength of \eqref{eq:alpha_plus} simplifies to
+Therefore, the transition strength of~\eqref{eq:alpha_plus} simplifies to
 \begin{equation}
 \alpha_{+,\text{bag}} = \frac{4 \Delta V}{3 w_+},
 \label{eq:alpha_plus_bag}
 \end{equation}
-and the transition strength at nucleation temperature from \eqref{eq:alpha_n} simplifies to
+and the transition strength at nucleation temperature from~\eqref{eq:alpha_n} simplifies to
 \begin{equation}
 \alpha_{n,\text{bag}} = \frac{4 \Delta V}{3 w_n}.
 \label{eq:alpha_n_bag}

tex/conclusion.tex

@@ -4,14 +4,16 @@
 and their gravitational wave spectra based on the Sound Shell Model,
 including the conversion to the gravitational spectra today.
 
-PTtools was tested with the constant sound speed model due to the availability of reference data by Giese et al. \cite{giese_2021},
+PTtools was tested with the constant sound speed model due to the availability of reference data by Giese et al.~\cite{giese_2021},
 and it was shown to work reliably for a broad range of the combinations of the sound speeds $c_{s,s}$ and $c_{s,b}$, the phase transition strength $\alpha_n$ and the wall speed $v_\text{wall}$.
 The PTtools fluid velocity profile solver has been updated beyond that of the available references in such a way
 that it supports arbitrary particle physics models with a temperature-dependent sound speed $c_s(T,\phi)$,
 as long as they provide $V_s(T,\phi), V_b(T,\phi)$ and two of $g_p(T,\phi), g_e(T,\phi), g_s(T,\phi)$ or two of $p(T,\phi), e(T,\phi), s(T,\phi)$, but preferably all three for numerical precision.
 This enables the easy integration of models developed by other researchers,
 and therefore the comparison of the gravitational wave spectra of these models.
-This is also to the author's best knowledge the first time that phase transitions in the early universe have been simulated with temperature-dependent speeds of sound without resorting to time-consuming 3D simulations.
+This is also to the author's best knowledge the first time
+that phase transitions in the early universe have been simulated with temperature-dependent speeds of sound
+without resorting to time-consuming 3D simulations.
 
 The numerical results of the fluid shell solver have been demonstrated to be consistent with a reference
 for a broad range of parameters.
@@ -31,15 +33,20 @@
 the effects of the phase transition parameters on the resulting gravitational wave spectrum.
 
 This master's thesis lays the groundwork for the author's PhD thesis on the topic.
-Now there is a framework for determining the gravitational wave spectrum from the phase transition parameters, including temperature-dependent speed of sound.
-Determining the parameters of a phase transition from LISA data will require solving the inverse problem: what are the phase transition parameters based on the gravitational wave spectrum?
+Now there is a framework for determining the gravitational wave spectrum from the phase transition parameters,
+including temperature-dependent speed of sound.
+Determining the parameters of a phase transition from LISA data will require solving the inverse problem:
+what are the phase transition parameters based on the gravitational wave spectrum?
 This will require novel tools such as machine learning or Markov chain Monte Carlo simulations.
-There is also potential for integrating PTtools with the existing web-based simulation utility PTPlot \cites{ptplot}{hindmarsh_shape_2017}{caprini_detecting_2020} to create a comprehensive but easy-to-use web-based utility for researchers to simulate phase transitions with various parameters.
-Further possibilities for extending PTtools itself include integrating the low frequency spectral density function by Giombi et al. \cite[eq. 3.6]{giombi_cs_2024}.
+There is also potential for integrating PTtools with the existing web-based simulation utility
+PTPlot~\cites{ptplot}{hindmarsh_shape_2017}{caprini_detecting_2020}
+to create a comprehensive but easy-to-use web-based utility for researchers to simulate phase transitions with various parameters.
+Further possibilities for extending PTtools itself include integrating
+the low frequency spectral density function by Giombi et al.~\cite[eq. 3.6]{giombi_cs_2024}.
 These topics will be investigated in the author's PhD studies.
 
-PTtools will be published as open source soon after the publication of this thesis on GitHub at \cite{pttools}.
-This will enable the research community to integrate their various particle physics models,
+PTtools is available as open source on GitHub at~\cite{pttools}.
+This enables the research community to integrate their various particle physics models,
 and to simulate the resulting gravitational wave spectra.
 This will provide the LISA simulation pipeline with various waveforms of interest,
 and if one of them is eventually found in LISA data,

tex/const_cs.tex

@@ -22,7 +22,7 @@ \section{The constant sound speed model}
 The parameters $\mu_\pm$ are determined by the sound speeds in the symmetric and broken phases with
 \cites[eq. 16]{giese_2021}[eq. 39]{giese_2020}
 \begin{equation}
-\mu_\pm \equiv 1 + \frac{1}{c_{s\pm}^2},
+\mu_\pm \equiv 1 + \frac{1}{c_{s\pm}^2}.
 \end{equation}
 This model reduces to the bag model,
 when $c_{s+}^2 = c_{s-}^2 = \frac{1}{3}$
@@ -42,7 +42,7 @@ \section{The constant sound speed model}
 \begin{equation}
 e_\pm = (\mu_\pm - 1) a_\pm \left( \frac{T}{T_0} \right)^{\mu_\pm - 4} T^4 + V_\pm.
 \end{equation}
-The equation for enthalpy density \eqref{eq:w_const_cs} can be inverted to give the temperature as a function of enthalpy
+The equation for enthalpy density~\eqref{eq:w_const_cs} can be inverted to give the temperature as a function of enthalpy
 \begin{equation}
 T_\pm = \left( \frac{w}{\mu_\pm a_\pm T_0^4} \right)^\frac{1}{\mu_\pm} T_0.
 \end{equation}
@@ -54,58 +54,58 @@ \section{The constant sound speed model}
 \end{align}
 
 
-The trace anomaly of eq. \eqref{eq:theta} simplifies to
+The trace anomaly of eq.~\eqref{eq:theta} simplifies to
 \begin{align}
 \theta_\pm(w)
 &= \left( \frac{\mu_\pm}{4} - 1 \right) a_\pm \left( \frac{T}{T_0} \right)^{\mu_\pm - 4} T^4 + V_\pm \\
 &= \left( \frac{1}{4} - \frac{1}{\mu_\pm} \right) w_\pm + V_\pm.
 \label{eq:theta_const_cs}
 \end{align}
-The transition strength of eq. \eqref{eq:alpha_plus} simplifies to
+The transition strength of eq.~\eqref{eq:alpha_plus} simplifies to
 \begin{equation}
 \alpha_+ = \frac{1}{3} \left( 1 - \frac{4}{\mu_+} \right) - \frac{1}{3} \left(1 - \frac{4}{\mu_-} \right) \frac{w_-}{w_+} + \alpha_{+,\text{bag}}.
 \label{eq:alpha_plus_const_cs}
 \end{equation}
-The transition strength at nucleation temperature of \eqref{eq:alpha_n} is correspondingly
+The transition strength at nucleation temperature of eq.~\eqref{eq:alpha_n} is correspondingly
 \begin{align}
 \alpha_n &= \frac{1}{3} \left( 1 - \frac{4}{\mu_+} \right) - \frac{1}{3} \left(1 - \frac{4}{\mu_-} \right) \frac{w(T_n, \phi_-)}{w_n} + \alpha_{n,\text{bag}} \\
 &= \frac{4}{3} \left( \frac{1}{\mu_-} - \frac{1}{\mu_+} \right) + \frac{1}{3} \left( 1 - \frac{4}{\mu_-} \right)
 \left( 1 - \frac{\mu_- a_-}{\mu_+ a_+} \left( \frac{T_n}{T_0} \right)^{\mu_- - \mu_+} \right) + \alpha_{n,\text{bag}}.
 \label{eq:alpha_n_const_cs}
 \end{align}
-When $\mu_- = 4$, $\alpha_n$ of eq. \eqref{eq:alpha_n_const_cs} becomes independent of $w_-$, resulting in
+When $\mu_- = 4$, $\alpha_n$ of eq.~\eqref{eq:alpha_n_const_cs} becomes independent of $w_-$, resulting in
 \begin{equation}
 w_n = \frac{4 (V_+ - V_-)}{3 \alpha_n + \frac{4}{\mu_+} - 1}.
 \label{eq:wn_const_cs_mu4}
 \end{equation}
 In this case we can obtain $v_{CJ}(\alpha_n, c_{s+})$ analytically.
-First we insert $\alpha_n$ to eq. \eqref{eq:wn_const_cs_mu4} to get $w_n$,
-and then insert $w_n$ to eq. \eqref{eq:alpha_plus_const_cs} to get $\alpha_+$,
+First we insert $\alpha_n$ to eq.~\eqref{eq:wn_const_cs_mu4} to get $w_n$,
+and then insert $w_n$ to eq.~\eqref{eq:alpha_plus_const_cs} to get $\alpha_+$,
 since in this case $\alpha_+$ is independent of $w_-$.
-Then we can insert $c_{s-}$ and $\alpha_+$ to eq. \eqref{eq:v_tilde_plus} to get $\tilde{v}_+ = v_{CJ}$.
+Then we can insert $c_{s-}$ and $\alpha_+$ to eq.~\eqref{eq:v_tilde_plus} to get $\tilde{v}_+ = v_{CJ}$.
 
-The pseudotrace of eq. \eqref{eq:theta_bar} is given by
+The pseudotrace of eq.~\eqref{eq:theta_bar} is given by
 \begin{align}
 \bar{\Theta} = (\mu_\pm - \mu_-) a \left(\frac{T}{T_0}\right)^{\mu_\pm - 4} T_0^4 + \mu_- V_\pm.
 \end{align}
-Therefore the corresponding transition strengths of eq. \eqref{eq:alpha_theta_bar_plus} and \eqref{eq:alpha_theta_bar_n} become
+Therefore the corresponding transition strengths of eq.~\eqref{eq:alpha_theta_bar_plus} and~\eqref{eq:alpha_theta_bar_n} become
 \begin{align}
 \alpha_{\bar{\Theta}_+} &= \frac{1}{3} \left(1 - \frac{\mu_-}{\mu_+}\right) + \frac{\mu_-}{4} \alpha_{+,\text{bag}}, \\
 \alpha_{\bar{\Theta}_n} &= \frac{1}{3} \left(1 - \frac{\mu_-}{\mu_+}\right) + \frac{\mu_-}{4} \alpha_{n,\text{bag}}.
 \end{align}
 These are equal for detonations regardless of the wall speed or the speeds of sound,
 since $\alpha_{+,\text{bag}} = \alpha_{n,\text{bag}}$ for detonations.
 
-If we set $T_0 = T_c$, eq. \eqref{eq:critical_temp} results in
+If we set $T_0 = T_c$, eq.~\eqref{eq:critical_temp} results in
 \begin{equation}
 \Delta V = (a_+ - a_-) T_c^4.
 \end{equation}
-Using this the ``bag'' part in eq. \eqref{eq:alpha_plus_const_cs} and \eqref{eq:alpha_n_const_cs} becomes
+Using this the ``bag'' part in eq.~\eqref{eq:alpha_plus_const_cs} and~\eqref{eq:alpha_n_const_cs} becomes
 \begin{align}
 \alpha_{+,\text{bag}} &= \frac{4}{3 \mu_+} \left( 1 - \frac{a_-}{a_+} \right) \left( \frac{T_+}{T_c} \right)^{-\mu_+}, \\
 \alpha_{n,\text{bag}} &= \frac{4}{3 \mu_+} \left( 1 - \frac{a_-}{a_+} \right) \left( \frac{T_n}{T_c} \right)^{-\mu_+}.
 \end{align}
-Using this we can reorder eq. \eqref{eq:alpha_n_const_cs} to
+Using this we can reorder eq.~\eqref{eq:alpha_n_const_cs} to
 \begin{equation}
 \frac{a_-}{a_+} = \frac{4 + \left( \mu_+ - 4 - 3 \mu_+ \alpha_n \right) \left(\frac{T_n}{T_c}\right)^{\mu_+}}{4 + \left( \mu_- - 4 \right) \left(\frac{T_n}{T_c}\right)^{\mu_-}}.
 \end{equation}
@@ -119,20 +119,20 @@ \section{The constant sound speed model}
 Even though we have used $T_0 = T_c$ in the intermediate steps, this restriction is general.
 \fi
 
-Approximating $T_+ \approx T_-$ results in eq. \eqref{eq:cs2_compact} being approximated as
+Approximating $T_+ \approx T_-$ results in eq.~\eqref{eq:cs2_compact} being approximated as
 \begin{equation}
 c_{s,b}^2
 \equiv \frac{dp_b}{de_b}
 = \frac{dp_b/dT}{de_b/dT} \big|_{T_+ \approx T_-}
 \approx \frac{\delta p_-}{\delta e_-}.
 \label{eq:const_cs_approx}
 \end{equation}
-In the case of the constant sound speed model $c_{s}$ is independent of temperature, and therefore this approximation is exact.
-With this approximation and the use of eq. \eqref{eq:delta_relation} the second junction condition of eq. \eqref{eq:junction_conditions_simplified} becomes
+In the case of the constant sound speed model, $c_{s}$ is independent of temperature, and therefore this approximation is exact.
+With this approximation and the use of eq.~\eqref{eq:delta_relation} the second junction condition of eq.~\eqref{eq:junction_conditions_simplified} becomes
 \begin{equation}
 \delta p_- \left(1 - \frac{\tilde{v}_+ \tilde{v}_-}{c_{s,b}^2} \right) = \tilde{v}_+ \tilde{v}_- De_+ - Dp_+.
 \end{equation}
-Using this, eq. \eqref{eq:delta_relation} and eq. \eqref{eq:enthalpy_sum} on the first junction condition of \eqref{eq:junction_conditions_simplified} results in
+Using this, eq.~\eqref{eq:delta_relation} and eq.~\eqref{eq:enthalpy_sum} on the first junction condition of~\eqref{eq:junction_conditions_simplified} results in
 \begin{align}
 \frac{\tilde{v}_+}{\tilde{v}_-}
 &\approx \frac{
@@ -145,10 +145,10 @@ \section{The constant sound speed model}
 \left( \frac{\tilde{v}_+ \tilde{v}_-}{c_{s,b}^2} - 1 \right) + 3\tilde{v}_+ \tilde{v}_- \alpha_{\bar{\Theta}_+}}.
 \label{eq:const_cs_vp_vm}
 \end{align}
-The only approximation in this calculation is that of eq. \eqref{eq:const_cs_approx}.
+The only approximation in this calculation is that of eq.~\eqref{eq:const_cs_approx}.
 As it is exact for the constant sound speed model,
-eq. \eqref{eq:const_cs_vp_vm} is exact for the constant sound speed model.
-Eq. \eqref{eq:const_cs_vp_vm} can be rearranged to
+eq.~\eqref{eq:const_cs_vp_vm} is exact for the constant sound speed model.
+Eq.~\eqref{eq:const_cs_vp_vm} can be rearranged to
 \begin{equation}
 \frac{\tilde{v}_+}{c_{s,b}^2} \tilde{v}_-^2
 + \left(3 \alpha_{\bar{\Theta}_+} - 1 - \tilde{v}_+^2 \left(\frac{1}{c_{s,b}^2} + 3 \alpha_{\bar{\Theta}_+} \right) \right) \tilde{v}_-
@@ -157,16 +157,16 @@ \section{The constant sound speed model}
 \end{equation}
 which is a basic second-order equation for $\tilde{v}_- ( \tilde{v}_+, \alpha_{\bar{\Theta}_+} )$.
 Since $\tilde{v}_+ = v_w$ for all detonations, and $\alpha_{\bar{\Theta}_+} = \alpha_{\bar{\Theta}_n}$ for all detonations in the constant sound speed model, this gives $\tilde{v}_- ( v_w, \alpha_{\bar{\Theta}_n} )$.
-% And since \eqref{eq:alpha_n_const_cs} gives $\alpha_n(\mu_+, \mu_-, \alpha_{n,\text{bag}})$, we have $\tilde{v}_- (v_w, ...)$
-Eq. \eqref{eq:const_cs_vp_vm} can also be rearranged to
+% And since~\eqref{eq:alpha_n_const_cs} gives $\alpha_n(\mu_+, \mu_-, \alpha_{n,\text{bag}})$, we have $\tilde{v}_- (v_w, ...)$
+Eq.~\eqref{eq:const_cs_vp_vm} can also be rearranged to
 \begin{equation}
 \left( \frac{1}{c_{s,b}^2} + 3\alpha_{\bar{\Theta}_+} \right) \tilde{v}_+^2
 - \left( \frac{1}{\tilde{v}_-} + \frac{\tilde{v}_-}{c_{s,b}^2} \right) \tilde{v}_+
 + 1 - 3\alpha_{\bar{\Theta}_+}
 = 0.
 \end{equation}
-This gives the Chapman-Jouguet speed $\tilde{v}_+ \left( \tilde{v}_- = c_{s,b} \right)$ as
-\cite[eq. 55]{giese_2020}
+This gives the Chapman-Jouguet speed $\tilde{v}_+ \left( \tilde{v}_- = c_{s,b} \right)$
+as~\cite[eq. 55]{giese_2020}
 \begin{equation}
 v_{CJ} = \frac{ 1 + \sqrt{ 3\alpha_{\bar{\Theta}_+} ( 1 - c_{s,b}^2 + 3 c_{s,b}^2 \alpha_{\bar{\Theta}_+} ) }}{ \frac{1}{c_{s,b}} + 3 c_{s,b} \alpha_{\bar{\Theta}_+}}.
 \end{equation}