Early Universe Nucleosynthesis

[Mordred]

All good, it took me several months studying various literature directly relating to CKMS for me to finally fill in the blanks and be comfortable working with it. It has been one of my goals in this thread. (Still is but now that I figured out how the cross sections connect to to the CKMS for both left and right hand particles. I can now look at the supersymmetric partners.

[Baron d'Holbach]

4 hours ago, Mordred said:

left and right hand particles

Have you looked into cosmic microwave background radiation and how it affect it? It may have influenced the generation and the amplification of chiral asymmetry and may have played a role in the emergence of chirality overall.

[Mordred]

Leptogenesis and baryogenesis would occur at the initial electroweak symmetry breaking stages prior to the dark ages where the mean free path of photons due to overall density is less than 10^-30 metres. The CMB data would unlikely be able to preserve any evidence as the expansion and slow roll stages of inflation would cause supercooling followed btmy reheating.

 However I'm not trying to solve either leptogenesis and baryogenesis. I already know the cross scatterings show that the right neutrino mixing angles would be insufficient in quantity via the Higgs seesaw to account for either That possibility is already well researched.

10 minutes ago, Baron d'Holbach said:

Have you looked into cosmic microwave background radiation and how it affect it? It may have influenced the generation and the amplification of chiral asymmetry and may have played a role in the emergence of chirality overall.

However there is current research studying neutrino oscillations itself that may or may not provide insight to the above.

Edited May 22, 2023 by Mordred

[Baron d'Holbach]

5 hours ago, Mordred said:

supersymmetric partners.

Why are you looking into this? It technically do not exist? Or you just want a philosophical footing? 

[Mordred]

To better understand the Weinberg mixing angles with regards to the CKMS matrix and to further examine the aspects of the seesaw mechanism of the Higgs field. Assuming supersymmetry though you would have supersymmetric Higgs partners as well. Supersymmetry though hasn't been disproven yet and is still viable. However our colliders are still too low an energy level to produce a supersymmetric particle. Were on the minimal border line however.

 From what I see the supersymmetric partners do not work in the current CKMS matrix so you would need a different matrix to account for them. That is what I'm confirming.

On 5/21/2023 at 8:44 PM, Baron d'Holbach said:

Why are you looking into this? It technically do not exist? Or you just want a philosophical footing? 

 

I was correct you need a super-CKMS matrix for supersymmetry. Details here

https://arxiv.org/pdf/0810.1613.pdfc

 

Bose Einstein QFT format.

\[|\vec{k_1}\vec{k_2}\rangle\hat{a}^\dagger(\vec{k_1})\hat{a}^\dagger(\vec{k_2})|0\rangle\]

\[\Rightarrow |\vec{k_1}\vec{k_2}\rangle= |\vec{k_2}\vec{k_1}\rangle\]

number operator

\[\hat{N}=\hat{a}^\dagger(\vec{k})\hat{a}\vec{k})\]

Hamilton operator

\[\hat{H}=\int d^3k\omega_k[\hat{N}(\vec{k})+\frac{1}{2}]\]

momentum of field

\[\hat{P}=\int d^3k\vec{k}[\hat{N}(\vec{k})+\frac{1}{2}]\]

renormlized Hamilton

\[\hat{H_r}=\int d^3 k\omega_k\hat{a}^\dagger(\vec{k})\hat{a}(\vec{k})\]

 

Edited March 26 by Mordred

[Mordred]

Higgs again.

\[m\overline{\Psi}\Psi=(m\overline{\Psi_l}\Psi_r+\overline{\Psi_r}\Psi)\]

\[\mathcal{L}=(D_\mu\Phi^\dagger)(D_\mu\Phi)-V(\Phi^\dagger\Phi)\]

4 effective degrees of freedom doublet complex scalar field.

with 

\[D_\mu\Phi=(\partial_\mu+igW_\mu-\frac{i}{2}\acute{g}B_\mu)\Phi\]\

\[V(\Phi^\dagger\Phi)=-\mu^2\Phi^\dagger\Phi+\frac{1}{2}\lambda(\Phi^\dagger\Phi)^2,\mu^2>0\]

in Unitary gauge

\[\mathcal{L}=\frac{\lambda}{4}v^4\]

\[+\frac{1}{2}\partial_\mu H \partial^\mu H-\lambda v^2H^2+\frac{\lambda}{\sqrt{2}}vH^3+\frac{\lambda}{8}H^4\]

\[+\frac{1}{4}(v+(\frac{1}{2}H)^2(W_mu^1W_\mu^2W_\mu^3B_\mu)\begin{pmatrix}g^2&0&0&0\\0&g^2&0&0\\0&0&g^2&g\acute{g}\\0&0&\acute{g}g&\acute{g}^2 \end{pmatrix}\begin{pmatrix}W^{1\mu}\\W^{2\mu}\\W^{3\mu}\\B^\mu\end{pmatrix}\]

Right hand neutrino singlet needs charge conjugate for Majorana mass term (singlet requirement)

\[\Psi^c=C\overline{\Psi}^T\]

charge conjugate spinor

\[C=i\gamma^2\gamma^0\] 

Chirality

\[P_L\Psi_R^C=\Psi_R\]

mass term requires

\[\overline\Psi^C\Psi\] grants gauge invariance for singlets only.

\[\mathcal{L}_{v.mass}=hv_{ij}\overline{I}_{Li}V_{Rj}\Phi+\frac{1}{2}M_{ij}\overline{V_{ri}}V_{rj}+h.c\]

Higgs expectation value turns the Higgs coupling matrix into the Dirac mass matrix. Majorana mass matrix eugenvalues can be much higher than the Dirac mass.

diagonal of

\[\Psi^L,\Psi_R\] leads to three light modes v_i with mass matrix

\[m_v=-MD^{-1}M_D^T\]

MajorN mass in typical GUT 

\[M\propto10^{15},,GeV\]

further details on Majorana mass matrix

https://arxiv.org/pdf/1307.0988.pdf

https://arxiv.org/pdf/hep-ph/9702253.pdf

 

 

 

Edited May 23, 2023 by Mordred

[Mordred]

Sterile Neutrino related research papers

Next decade of sterile neutrino studies

by Alexey Boyarsky, Dmytro Iakubovskyi, Oleg Ruchayskiy

https://arxiv.org/pdf/1306.4954.pdf

Detection of An Unidentified Emission Line in the Stacked X-ray spectrum of Galaxy Clusters

Esra Bulbul, Maxim Markevitch, Adam Foster, Randall K. Smith, Michael Loewenstein, Scott W. Randall

https://arxiv.org/abs/1402.2301

Neutrino Masses, Mixing, and Oscillations

Revised October 2021 by M.C. Gonzalez-Garcia (YITP, Stony Brook; ICREA, Barcelona; ICC, U. of Barcelona) and M. Yokoyama (UTokyo; Kavli IPMU (WPI), UTokyo).

https://pdg.lbl.gov/2022/reviews/rpp2022-rev-neutrino-mixing.pdf

 

Edited May 24, 2023 by Mordred

[Mordred]

seesaw mechanism

righthand neutrino states with Higgs coupling

\[f^v \varepsilon_{ab}\overline{L}^aH^bV_r\]

which gives rise to Dirac mass term

\[M_D(\overline{V_L}V_R+\overline{V}_RV_L\]

Majorona mass terms

\[M_{m1}\overline{V_L}V^c_L+M_{2}M^{-c}_RV_R+c.c\]

\[\begin{pmatrix}\overline{V_L}\\\overline{V^c_R}\end{pmatrix}\begin{pmatrix}M_{m1}&M_D\\M_D&M_{M12}\end{pmatrix}(V^c_LV_R)\]

eugenvalues

\[\lambda^2=(M_{m1}+M_{M2})\lambda(M_{M1}M_{M2}-M_D^2)=0\]

solution

\[\lambda=\frac{(M_{M1}+M_{M2}\pm\sqrt{M_{(M1}-M_{M2}^2+4M_D^2}}{2}\]

as one eugenvalue increases the other decreases.

set 

\[M_{M1}=0,,,,M_{M2}>>M_D\]

gives

\[\lambda=M_{M2}(\frac{1\pm\sqrt{1+4}(\frac{M_D}{M_{M2}^2})}{2})\]

\[\lambda_1\approx M_{M2},\lambda_2\approx \frac{M^2_D}{M_M^2}\]

Edited May 25, 2023 by Mordred

[Mordred]

Reminder notes

Curl of a vector field definition

if vector F equals P,Q,R as a vector field in R^3 and \[P_x,Q_y, R_z\] all exists the the curl F is defined as

curl \[\vec{F}=(R_y-Q_z)\hat{i}+(P_z-R_x)\hat{J}+(Q_x-P_y)\hat{k}=(\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z})\hat{i}+(\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x})\hat{J}+(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})\hat{k}\]

the curl of a vector is a vector field in contrast to divergence given as

\[div \vec{F}=\vec{\nabla}\cdot\vec{F}\]

\[\vec{\nabla}x\vec{F}\]

\[\begin{pmatrix}\hat{i}&\hat{j}&\hat{k}\\\frac{\partial}{\partial x}&\frac{\partial}{\partial y}&\frac{\partial}{\partial z}\\P&Q&R\end{pmatrix}\] 

with determinant loosely defined as

\[(R_y-Q_z)\hat{i}-(R_x-P_z)\hat{j}-(Q_z-P_y)\hat{j}=(R_y-Q_z)\hat{i}+(R_x-P_z)\hat{j}+(Q_z-P_y)\hat{j}=curl \vec{F}\]

above definitions from

https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/16%3A_Vector_Calculus/16.05%3A_Divergence_and_Curl

pursuant next study gravity is divergent free on one loop integrals but divergent on 2 loop

 

Edited May 28, 2023 by Mordred

[Mordred]

future references with regards to Einstein-Hilbert action one loop integrals and two loop integrals.

https://arxiv.org/pdf/1706.02622.pdf

https://cds.cern.ch/record/261104/files/CM-P00049196.pdf

https://arxiv.org/abs/1207.2302

https://arxiv.org/pdf/hep-th/9605057.pdf

Quantum geometrodynamics

https://arxiv.org/abs/0812.0295

loop quantum gravity

https://arxiv.org/abs/1201.4598

https://www.cpt.univ-mrs.fr/~rovelli/IntroductionLQG.pdf

Edited May 31, 2023 by Mordred

[Mordred]

BRST quantization aka Faadev Popoff Ghosts

https://saalburg.aei.mpg.de/wp-content/uploads/sites/25/2017/03/henneaux.pdf

https://arxiv.org/pdf/1407.7256.pdf

appears to be the most common method of renormalization with regards to quantum gravity. 

Edited June 7, 2023 by Mordred

[Mordred]

start of nucleosynthesis

\[\rho_r c^2=\frac{3}{32\pi}\frac{c^2}{G_N}t^{-2}\]

\[\rho_r c^2=\frac{3}{32\pi}(\rho c^2)_{PL}(\frac{t_{PL}}{t})\]

\[K_b T\simeq 0.46 E_{PL}(\frac{t}{t_{PL}})^{-1/2}\]

\[t_s\simeq \frac{10^{20}}{[T(K)]^2}\]\[\simeq (T_{bbn}=10^9 K)\]

roughly \(10^2) seconds after BB

Edited June 10, 2023 by Mordred

[Mordred]

photon propogator

\[\frac{i}{k^2}[-g^{\mu\nu}+(1-\zeta)\frac{k^\mu k\nu}{k^2}]\]

in Feymann gauge \(\zeta=1\) gives

\[-\frac{i}{k^2}g^{\mu\nu}\]

polarization states of photon

\[\epsilon_1=\begin{pmatrix}0\\1\\0\\0\end{pmatrix}\]

\[\epsilon_2=\begin{pmatrix}0\\0\\1\\0\end{pmatrix}\]

normalization given by

\(\epsilon_1 \cdot \epsilon_2=g^{\mu\nu}\)

Electron/positron propogator

\[\frac{i(\gamma^\mu q_\mu+m)}{q^2-m^2}\]

delta function

\((2\pi)^4\varphi  ( p_1-p_2-q)\)

Edited June 11, 2023 by Mordred

[Mordred]

Slow Roll single scalar field perturbation 

\[[\delta\frac{\tilde{p}}{\rho}]^2=\frac{k^3}{2\pi^3}\int d^3 xe^{i\vec{k}\cdot \vec{x}}\langle \frac{\partial \rho}{\rho}\vec{x},t \frac{\partial \rho}{\rho}\vec{O},t\rangle\]

\[[\delta\tilde{t}(\vec{k})]^2=\frac{k^3}{(2\pi)^3}\int d^3xe^{i\vec{k}\cdot\vec{x}}\langle \partial t\vec{x}\partial{t}\vec{O}\rangle\]

Edited June 14, 2023 by Mordred

[Mordred]

Building the full Pontecorvo-Maki-Nakagawa-Sakata matrix from six independent Majorana-type phases

https://cds.cern.ch/record/1127373/files/GetPDFServlet.pdf

further examining the following from the article

\(\frac{N_b}{N_\gamma}=(6.1^{+0.3}_{-0.2})*10^{-10}\)

Quote

In this framework a CP asymmetry is generated through out-of-equilibrium L-violating decays of heavy Majorana neutrinos [38] leading to a lepton asymmetry which, in the presence of ðB þ LÞ-violating but ðB LÞ-conserving sphaleron processes [53], produces a baryon asymmetry. In the single flavor approach, with three singlet heavy neutrinos Ni, thermal leptogenesis is insensitive to the CP-violating phases appearing in the PMNS matrix. In this case there is complete decoupling among the phases responsible for CP violation at low energies and those responsible for leptogenesis [54,55].

hrrm this seems to imply this Cern paper considers right hand neutrinos accounting for leptogenesis.

Edited June 22, 2023 by Mordred

[Mordred]

Bump going to continue with this project

[Mordred]

For reference an extremely handy Feymann rules listing

https://porthos.tecnico.ulisboa.pt/CTQFT/files/SM-FeynmanRules.pdf

 

[Mordred]

Just pulling a link to an older thread with some articles and references to use for this thread.

[Mordred]

\[R^{\mu'}_{\phantom{\mu'}\nu'\alpha'\beta'}=\dfrac{\partial x^{\mu'}}{\partial x^\mu}\dfrac{\partial x^\nu}{\partial x^{\nu'}}\dfrac{\partial x^\alpha}{\partial x^{\alpha'}}\dfrac{\partial x^\beta}{\partial x^{\beta'}}R^\mu_{\phantom{\mu}\nu\alpha\beta}\]

[Mordred]

Christoffels for the FLRW metric in spherical coordinates.

\[ds^2=-c(dt^2)+\frac{a(t)}{1-kr^2}dr^2+a^2(t)r^2 d\theta^2+a^2(t)r^2sin^2d\phi\]

\[g_{\mu\nu}=\begin{pmatrix}-1&0&0&0\\0&\frac{a^2}{1-kr^2}&0&0\\0&0&a^2 r^2&0\\0&0&0&a^2r^2sin^2\theta \end{pmatrix}\]

\[\Gamma^0_{\mu\nu}=\begin{pmatrix}0&0&0&0\\0&\frac{a}{1-(kr^2)}&0&0\\0&0&a^2r^2&0\\0&0&0&a^2r^2sin^2\theta \end{pmatrix}\]

\[\Gamma^1_{\mu\nu}=\begin{pmatrix}0&\frac{\dot{a}}{ca}&0&0\\\frac{\dot{a}}{ca}&\frac{a\dot{a}}{c(1-kr^2)}&0&0\\0&0&\frac{1}{c}a\dot{a}r^2&0\\0&0&0&\frac{1}{c}a\dot{a}sin^2\theta \end{pmatrix}\]

\[\Gamma^2_{\mu\nu}=\begin{pmatrix}0&0&\frac{\dot{a}}{ca}&0\\0&0&\frac{1}{r}&0\\\frac{\dot{a}}{ca}&\frac{1}{r}&0&0\\0&0&0&-sin\theta cos\theta \end{pmatrix}\]

\[\Gamma^3_{\mu\nu}=\begin{pmatrix}0&0&0&\frac{\dot{a}}{ca}\\0&0&0&\frac{1}{r}\\0&0&0&cot\theta\\\frac{\dot{a}}{c}&\frac{1}{r}&cot\theta&0\end{pmatrix}\]

\(\dot{a}\) is the velocity of the scale factor if you see two dots its acceleration in time derivatives. K=curvature term

Newton limit geodesic

\[\frac{d^r}{dt^2}=-c^2\Gamma^1_{00}\]

Christoffel Newton limit

\[\Gamma^1_{00}=\frac{GM}{c^2r^2}\]

Covariant derivative of a vector \(A^\lambda\)

\[\nabla_\mu A^\lambda=\partial_\mu A^\lambda+\Gamma_{\mu\nu}^\lambda A^\nu\]

Edited Tuesday at 12:50 PM by Mordred

[Mordred]

Palatini Higgs

Induced gravity scenario 

\[S=\int d^4x\sqrt{-g}[\frac{\xi h^2}{2}R-\frac{1}{2}(\partial h)^2-\frac{1}{4}h^4-\frac{1}{4}f_{\mu\nu}F^{\mu\nu}-\frac{g^2}{4}h^2B_\mu B^\mu-i\bar{\psi}{\not}\tiny\,\normalsize\partial\psi-\frac{\gamma}{\sqrt{2}}h\bar{\psi}{\psi}]\]

{\not}\tiny\,\normalsize\partial

scalar field h, vector field \(B_{\mu\nu}\) fermion field \(\psi\) above Abelion with standard \(B_{\mu}B^{\nu}\) kinetic terms \(F_{\mu\nu}F^{\mu\nu}\)

scalar field expectation value

\[G_{n,eff}\equiv\frac{1}{8\pi\xi h^2}\]

to keep \(G_{n,eff}\) well behave non-minimal coupling \(\xi\) is constrained to positive values for semi-positive definiteness of the scalar field kinetic term. shown by a field redefinition \(h^2\rightarrow h^2\xi\)

Einstein-Hilbert frame redefinition of the metric terms.

\(g_{\mu\nu}\rightarrow \Theta g_{\mu\nu},,\Theta \equiv \frac{F^2_\infty}{h^2},,,F_\infty\equiv\frac{m_P}{\sqrt{\xi}}\)

with rescaling of the vector and fermion fields

\(A_\mu \rightarrow \Theta^{-1/2},,,,\psi \rightarrow \Theta^{-3/4}\psi\)

\[S=\int d^4 x[\frac{M_p^2}{2}R-\frac{1}{2}m^2_P K(\Theta)(\partial\Theta)^2-\frac{\lambda}{4}F_{\mu\nu}F^{\mu\nu}-\frac{g^2}{4}F^2_\infty B_\mu B^{\mu}-i\bar{\psi} {\not}\tiny\,\normalsize\partial \psi-\frac{\gamma}{\sqrt{2}}T_\infty \bar{\psi}\psi]\]

contains non-canonical term for the \(\Theta\) with kinetic coefficient 

\[K(\Theta)\equiv\frac{1}{4|a|\Theta^2}\] 

quadrupole at \(\Theta=0)\) and a constant \[a\equiv\frac{\xi}{1+6\xi}<0\]

canonical field correction via field redefinition 

\[\Theta^{-1}=exp(\frac{2\sqrt{|a|\phi}}{M_P})\]

\[ S=\int dx^4 x\sqrt{-g}[\frac{M_P^2}{2}R-\frac{1}{2}(\partial\phi)^2-\frac{\lambda}{4}F^4_\infty-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{g^2}{4}F^2_\infty B_\mu B^\mu-i\bar{\psi}{\not}\tiny\,\normalsize\partial \psi+\frac{y}{\sqrt{2}}F_\infty\bar{\phi}\phi ]\]

 

https://arxiv.org/abs/1807.02376

 

Edited April 21 by Mordred

[Mordred]

\[{\small\begin{array}{|c|c|c|c|c|c|c|c|c|c|}\hline Field & \ell_L& \ell_R &v_L&U_L&d_L&U_R &D_R&\phi^+&\phi^0\\\hline T_3&- \frac{1}{2}&0&\frac{1}{2}&\frac{1}{2}&-\frac{1}{2}&0&0&\frac{1}{2}&-\frac{1}{2} \\\hline Y&-\frac{1}{2}&-1&-\frac{1}{2}&\frac{1}{6}&\frac{1}{6}& \frac{2}{3}&-\frac{1}{3}&\frac{1}{2}&\frac{1}{2}\\\hline Q&-1&-1&0&\frac{2}{3}&-\frac{1}{3}&\frac{2}{3}&-\frac{1}{3}&1&0\\\hline\end{array}}\]

 

Edited Sunday at 10:21 PM by Mordred

[Orion1]

On 4/4/2024 at 11:32 PM, Mordred said:

Christoffels for the FLRW metric in spherical coordinates.

[math]\color{blue}{\text{Your LaTex Karate has improved, what LaTex software are you using?, just some clarification on formal denotation.}}[/math]
[math]\;[/math]
[math]\color{blue}{g_{\mu \nu} \text{ and } g_{\alpha \beta} \text{ are formally denoted for the metric spacetime tensor in General Relativity.}}[/math]
[math]\color{blue}{G_{\mu \nu} \text{ and } G_{\alpha \beta} \text{ are formally denoted for the Einstein tensor in General Relativity.}}[/math]
[math]\;[/math]
[math]\color{blue}{\text{The Friedmann–Lemaître–Robertson–Walker FLRW metric:} \; (\text{ref. 1})}[/math]
[math]ds^2 = -c \; dt^2 + \frac{a\left(t \right)^2}{1 - k r^2} dr^2 + a\left(t \right)^2 r^2 d \theta^2 + a\left(t \right)^2 r^2 \sin^2 \theta \; d \phi^2[/math]
[math]\;[/math]
[math]\color{blue}{\text{The metric spacetime tensor in General Relativity for the FLRW metric:} \; (\text{ref. 3, sec. 3.2})}[/math]
[math]g_{\mu \nu} = \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & \frac{a\left(t \right)^2}{1 - k r^2} & 0 & 0 \\ 0 & 0 & a\left(t \right)^2 r^2 & 0 \\ 0 & 0 & 0 & a\left(t \right)^2 r^2 \sin^2 \theta \end{pmatrix}[/math]
[math]\;[/math]
[math]\color{blue}{\text{The Einstein tensor in General Relativity:} \; (\text{ref. 2, ref. 3, eq. 3.17})}[/math]
[math]G_{\mu \nu } = R_{\mu \nu } - \frac{1}{2} g_{\mu \nu } R[/math]
[math]\;[/math]
[math]\color{blue}{\text{Any discussions and/or peer reviews about this specific topic thread?}}[/math]
[math]\;[/math]
Reference:
Friedmann-Lemaître-Robertson-Walker metric: (ref. 1)
https://en.wikipedia.org/wiki/Friedmann-Lemaître-Robertson-Walker_metric
Wikipedia - Einstein tensor (ref. 2)
https://en.wikipedia.org/wiki/Einstein_tensor
Relativistic Cosmology - M. Pettini: (ref. 3)

https://people.ast.cam.ac.uk/~pettini/Intro Cosmology/Lecture03.pdf

Edited Tuesday at 07:46 AM by Orion1
source code correction...

[Mordred]

Hey  @Orion1 welcome back mate. No software I manually type in the latex. Lol thanks for the reminder to keep the metric tensor separate from the Einstein tensor lol

Edited Tuesday at 08:15 AM by Mordred

[Mordred]

On 4/4/2024 at 11:32 PM, Mordred said:

Christoffels for the FLRW metric in spherical coordinates.

 

ds2=−c(dt2)+a(t)1−kr2dr2+a2(t)r2dθ2+a2(t)r2sin2dϕ

 

 

gμν=⎛⎝⎜⎜⎜⎜−10000a21−kr20000a2r20000a2r2sin2θ⎞⎠⎟⎟⎟⎟

 

 

Γ0μν=⎛⎝⎜⎜⎜⎜00000a1−(kr2)0000a2r20000a2r2sin2θ⎞⎠⎟⎟⎟⎟

 

 

Γ1μν=⎛⎝⎜⎜⎜⎜⎜⎜0a˙ca00a˙caaa˙c(1−kr2)00001caa˙r200001caa˙sin2θ⎞⎠⎟⎟⎟⎟⎟⎟

 

 

Γ2μν=⎛⎝⎜⎜⎜⎜⎜00a˙ca0001r0a˙ca1r00000−sinθcosθ⎞⎠⎟⎟⎟⎟⎟

 

 

Γ3μν=⎛⎝⎜⎜⎜⎜⎜000a˙c0001r000cotθa˙ca1rcotθ0⎞⎠⎟⎟⎟⎟⎟

 

a˙ is the velocity of the scale factor if you see two dots its acceleration in time derivatives. K=curvature term

Newton limit geodesic

 

drdt2=−c2Γ100

 

Christoffel Newton limit

 

Γ100=GMc2r2

 

Covariant derivative of a vector Aλ

 

∇μAλ=∂μAλ+ΓλμνAν

 

Correction applied lol