# Ganesh's log

## Ideas to use hackmd

- `hackmd` using markdown to write
- text based
- markdown is like simplified `html` + `latex`
- able to add in equations, figures and table
- [https://www.markdownguide.org/basic-syntax/](https://www.markdownguide.org/basic-syntax/)

- use it as your daily logs and meeting logs

This `hackmd` page is shared. All of us can access, if there is any ideas/questions you can write here to record down and we can discuss in our meeting.

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- example of inserting a figure:


## Example of equations

- Inline equation example: $ds^2 = \sum_{\mu}\sum_{\nu} g_{\mu\nu} dr^{\mu} dr^{\nu} = g_{\mu\nu} dr^{\mu} dr^{\nu}$
- paragraph equation example, $$ds^2 = \sum_{\mu}\sum_{\nu} g_{\mu\nu} dr^{\mu} dr^{\nu} = g_{\mu\nu} dr^{\mu} dr^{\nu}$$

<hr>

# Summary of what we have discussed so far:
## Introduction to Cosmology
- Derivation of FLRW metric: $ds^2=dt^2-a^2(t)(\frac{dr^2}{1-kr^2}+r^2d\theta^2+r^2sin^2(\theta)d\phi^2)$, c=1. Where when $k=0$, $k=-1$ and $k=1$, the Universe have Flat, Hyperbolic and Spherical geometry.
- Redshift: $1+z=\frac{a(t_o)}{a(t_e)}=\frac{\lambda_o}{\lambda_e}$, where $t_o$ and $t_e$ representing time when the light signal emitted and observed respectively. $a$ and $\lambda$ representing scale factor and wavelength.
- First Friedmann equation: $\dot{a}=\frac{8\pi G}{3}\rho a^2-k$, can be expressed as $H=\frac{8\pi G}{3}\rho-\frac{k}{a^2}$.
- Second Friedmann equation (also known as acceleration equation): $\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}\big(\rho+3p\big)$, can also expressed as $H^2+\dot{H}=-\frac{4\pi G}{3}\big(\rho+3p\big)$.
- Fluid equation: $\dot{\rho}=-3H(\rho+p)$.
- From Thermodynamics, using Ideal gas law one can derive equation of state parameter, $\omega=\frac{1}{3}\frac{<\nu^2>}{c^2}$. Where $\omega=0$, $\omega=\frac{1}{3}$ and $\omega=-1$ for dust (non-relativistic matter), radiation (relativistic matter) and Cosmological constant dominated Universe respectively.Therefore, we have equation of state $p=\omega\rho$ where $p,\rho$ are pressure and energy density of the matter respectively.
- One can deduce general formula for the evolution of the energy density of the universe using Fluid equation as $\rho=\rho_0(\frac{a_0}{a})^{3(1+\omega)}$. One can draw a graph as follows:
From the graph above we knows that there was a time when both radiation and matter dominated Universe happened at the same time (transition period). Later we will try calculate the scale factor $a(t)$ and time when this happened.
# Kinematic properties of Fluids (29/12/2021)
## Kinematics Shear ($\sigma_{\mu\nu}$), Epansion ($\theta$) and Vorticity ($\omega_{\mu\nu}$)
- We have fluid 4-velocity in coordinate $x^\mu=(t,x^i)$, where Greek indices representing the spacetime coordinates and Latin indices representing spatial coordinates as $u^\mu=\frac{dx^\mu}{d\tau}$, where $\tau$ is the proper time.
- We can define fluid 4-acceleration as $a^\mu=u^\nu u^\mu_{;\nu}$, where ";" representing the covariant derivative.
- These two fundamental quantities, 4-velocity and 4-acceleration give rise to two conditions which are normalisation and orthogonal relation respectively, ${\bf{u}\cdot\bf{u}}=u^\mu u_\mu=-1$ and ${\bf{a}\cdot\bf{u}}=a^\mu u_\nu=0$.
- One can easily prove the identity $u^\mu u_{\mu;\nu}=0$ from the normalisation condition.
- The fluid 4-velocity can be expressed as $u^\mu=u^0(1, v^i)$ where $v^i$ is the fluid spatial velocity
- All the kinematic properties of the fluid can be expressed in terms of $\bf{u}$ and $\bf{a}$.To show this, one can introduce a displacement vector, $\xi^\mu$ as a 4-vector connecting two fluid element which are very close to each other as they move along their worldline but will not be geodesics.We have rate of change of displacement vector as $u^\mu\xi^\nu_{;\mu} but since displacement vector, $\bf{\xi}$ is Lie dragged along fluid 4-velocity $\bf{u}, we have $\mathcal{L}_{\bf{u}}{\bf{\xi}}=0$.Therefore, in the components form we have $u^\mu\xi^\nu_{;\mu}=\xi^\mu u^\nu_{;\mu}$, then we can define a rate of change of displacement vector as $\dot{\xi^\mu}:=u^\nu\xi^\mu_{;\nu}=\xi^\nu u^\mu_{;\nu}$.
- Any covariant rank 2 tensor can be decomposed into irreducible form, $W_{\mu\nu}=Au_\mu u_\nu+B_\mu u_\nu+u_\mu C_\nu+\frac{1}{3}W_{\alpha\beta}h^{\alpha\beta}h_{\mu\nu}+W_{<\mu\nu>}+h^\alpha_\mu h^\beta_\nu W_{[\alpha\beta]}$, where $A:=W_{\mu\nu}u^\mu u^\nu$, $B_\mu:=-h^\alpha_{\mu} W_{\alpha\beta}u^\beta$, $C_\nu:=-h^\alpha_\nu W_{\beta\alpha}u^\beta$, $W_{<\mu\nu>}:=h_\mu^\alpha h^\kappa_\nu W_{(\alpha\kappa)}-\frac{1}{3}W_{\alpha\kappa}h^{\alpha\kappa}h_{\mu\nu}$ and $h^\alpha_{\mu} h^\beta_{\nu} W_{[\alpha\beta]}:=h^\alpha_{[\mu} h^\beta_{\nu]} W_{\alpha\beta}$. Applying this irreducible decomposion to $u_\mu;\nu$, we have $\dot{\xi^\mu}:=\xi^\nu u_{\mu;\nu}=\xi^\nu(\omega_{\mu\nu}+\sigma_{\mu\nu}+\frac{1}{3}\theta h_{\mu\nu}-a_\mu u_\nu)$, where $h_{\mu\nu}:=g_{\mu\nu}+u_\mu u_\nu$ is the projection tensor.
- Therefore, any covariant derivative of covariant fluid 4-velocity can be experessed as $u_{\mu;\nu}=\omega_{\mu\nu}+\sigma_{\mu\nu}+\frac{1}{3}\theta h_{\mu\nu}-a_\mu u_\nu$, where here the irreducible tensorial part are **vorticity (or twist) tensor**, $\omega_{\mu\nu}:=h^\alpha_\nu h^\beta_\nu u_{[\alpha;\beta]}=u_{[\mu;\nu]}+a_{[\mu}u_{\nu]}$; **shear tensor**, $\sigma_{\mu\nu}:=u_{<\mu;\nu>}=u_{(\mu;\nu)}+a_{(\mu}u_{\nu)}-\frac{1}{3}\theta h_{\mu\nu}$ and **expansion scalar**, $\theta:=h^{\mu\nu}u_{\mu;\nu}=u^\mu_{;\mu}$.
- The following relations are holds for the kinematic tensors, $\omega_{(\mu\nu)}=\sigma_{[\mu\nu]}=0$, $\sigma^\mu_\mu=0$, $\omega_{\mu\nu}u^\nu=0=\sigma_{\mu\nu}u^\nu$,$\omega_{\mu\nu}h^{\mu\nu}=0$ and $\sigma_{\mu\nu}h^{\mu\nu}=0$.
- The information one can grasp from **vorticity tensor, $\omega_{\mu\nu}$** is that it tells about the rotation or the twist of the fluid elements, **shear tensor, $\sigma_{\mu\nu}$** describes the deformations and the eigenvalues of $\sigma^{\mu\nu}$ are the rate of change of the axial length of the fluid element, during the deformation the volume is preserved as the deformation takes place while the trace of $\sigma^{\mu\nu}$ is zero. The **expansion scalar, $\theta$** describes the change of volume while keeping the shape same.

## Evolution laws for the kinematic quantities (10/1/2022)
- We can apply the idea of torsionless noncommutative covariant derivative of covariant derivative vector field $u_\lambda$ to study the evolution of kinematic equation (vorticity, shear tensor and expansion scalar) described in the previous section. We have $u_{\lambda;[\nu\mu]}=u_{\lambda;\nu\mu}-u_{\lambda;\mu\nu}=R^\kappa_{\ \ \ \lambda\nu\mu}u_\kappa$ where $R^\kappa_{\ \ \ \lambda\nu\mu}u_\kappa=\partial_{[\nu}\Gamma^{\kappa}_{\lambda\mu]}-\Gamma^\kappa_{\alpha[\nu}\Gamma^\alpha_{\lambda\mu]}$ is Riemann tensor.
- One can rearrange the above mentioned equation as $u_{\lambda;\nu\mu}u^\mu=R^\kappa_{\ \ \ \lambda\nu\mu}u_\kappa u_\mu+u_{\lambda;\mu\nu}u^\mu=R^{\quad \kappa}_{\mu\nu\lambda}u^\mu u_\kappa+(u_{\lambda;\mu}u^\mu)_{;\nu}-u^\mu_{;\nu}u_{\lambda;\mu}$.
- We had $u_{\mu;\nu}=\omega_{\mu\nu}+\sigma_{\mu\nu}+\frac{1}{3}\theta h_{\mu\nu}-a_\mu u_\nu$,then $(\omega_{\lambda\nu}+\sigma_{\lambda\nu}+\frac{1}{3}\theta h_{\lambda\nu}-a_\lambda u_\nu)_{;\mu}u^\mu=R^{\quad \kappa}_{\mu\nu\lambda}u^\mu u_\kappa+(u_{\lambda;\mu}u^\mu)_{;\nu}-u^\mu_{;\nu}u_{\lambda;\mu}$
## Discussion on the LRS Bianchi-I Spacetime with viscosity (17/1/2022)
- We discuss some calculations relating the LRS Bianchi-I Spacetime with viscosity.
- Trying to share ideas on solving the equations.
- Discussing the ideas in the 1985 paper on Bianchi Type-II Cosmological Model with Viscous Fluid (https://arxiv.org/pdf/2105.03593.pdf) which can be applied in our studdies.

# Cosmological Perturbation Theory
## Summary (16/2/2022)
- We learn about coordinates transformation of a point $\bar{P}$ in background space $x^\alpha$ and two other coordinate systems $\hat{x}^\alpha$ and $\tilde{x}^\alpha$.
- We have discussed about relating the point $\bar{P}$ in background space to other coordinate system, $\hat{x}^\alpha$ and $\tilde{x}^\alpha$.
- We discussed how scalars, vectors and tensors are transform and then relating it to the perturbation respectively.
- We come into a conclusion that the difference in perturbation in two coordinate system can be related to Lie derivative of the background spacetime w.r.t a first order small quantity.
## Summary (28/2/2022)
- We have discused that the difference of perturbations of scalars, vectors and tensors in coordinates ($\tilde x$) and ($\hat{x}$) (sometimes the coordinates are called as gauge here in cosmological perturbation) can be expressed as Lie derivatives of background scalar, vectors and tensors respectively w.r.t small quantity ($\xi$). Note that background quantities are the one with a bar above e.g. $\bar{A^\alpha_\beta}$.
- We have discussed how the perturbation of the components of the a mixed tensor $\bar{A^\alpha_\beta}$ in background spacetime changed according to the gauge transformation.
- Finally, we discussed metric perturbation, $\delta g_{\mu\nu}$ by using perturbation of rank two tensors. We found the relation of the perturbed metric in one gauge ($\tilde{x}$) to the other gauge ($\hat{x}$), where the "hat" is often not written.
## Summary (14/3/2022)
- We discussed the transformation of the components of the perturbed metric w.r.t the gauge transformation.
- We also discussed the separation of the components of the perturbed metric into scalars, vectors and tensors.
- Metric perturbation can be devided into 3 parts, scalars ($A, B, D$ and $E$), Vectors ($B_i^V$, $E_i$) and Tensors ($E^T_{ij}$).
- 10 Degree of freedom of the Metric Perturbation can be devided into 4 scalars, 4 Vectors and 2 Tensors.
- Each of this scalars, vectors and tensors evolved independently (up to 1st order perturbations only). Amongst them, scalar perturbation is important for formation of structure in the Universe, vector perturbation coupled to the rotational velocity perturbation of the cosmic fluid, they tend to decay in expanding Universe and tensor perturbation are corresponding to the gravitational waves. A strong tensor perturbation have observable effect on anisotropy of CMB.
## Summary (28/3/2022)
- We described the perturbations we learnt before using Fourier space.
- In the first order perturbation all parts e.g scalars, vector and tensors evolves independently, therefore the corresponding Fourier parts evolves similarly.
- In a comoving coordinates $\overrightarrow{x}=(x^1, x^2, x^3)$, one can have comoving wave vectors $\overrightarrow{k}$, comoving wavenumber $k=|\overrightarrow{k}|$ and the wavelength $\lambda=\frac{2\pi}{k}$.
- The physical wavelength $\lambda_{phys}=a(t)\lambda$ where $a(t)$ is the scale factor. The physical wavelength expands with the expansion of the Universe.
## Summary (11/4/2022)
- We have continued our previous discussion to separation into scalars, vectors and tensors. We have the following $$B(\eta, \overrightarrow{x})=\Sigma_{\overrightarrow{k}}\frac{B_{\overrightarrow{k}}(\eta)}{k}e^{i\overrightarrow{k}\cdot{\overrightarrow{x}}}=\frac{1}{(2\pi)^{3/2}}\int_V\frac{B_{\overrightarrow{k}}(\eta)}{k}e^{i\overrightarrow{k}\cdot{\overrightarrow{x}}}d^3\overrightarrow{k},$$ $$E(\eta, \overrightarrow{x})=\Sigma_{\overrightarrow{k}}\frac{E_{\overrightarrow{k}}(\eta)}{k}e^{i\overrightarrow{k}\cdot{\overrightarrow{x}}}=\frac{1}{(2\pi)^{3/2}}\int_V\frac{E_{\overrightarrow{k}}(\eta)}{k}e^{i\overrightarrow{k}\cdot{\overrightarrow{x}}}d^3\overrightarrow{k}.$$
- Note that the Fourier sum can be changed to integral when the volume, $V$ approaches inifinity.
- Extending the smilar ideas as discussed before for separation of perturbation into scalars, vectors and tensors to Fourier space, we have the scalars, vectors and tensors component of the perturbation as follows in Fourier space $$\delta g_{\mu\nu}^S=a^2\begin{pmatrix}-2A & & & +iB\\& 2(-D+\frac{1}{3}E) & & \\& &2(-D+\frac{1}{3}E)& \\+iB& & & 2(-D-\frac{2}{3}E)\end{pmatrix},$$ $$\delta g_{\mu\nu}^v=a^2\begin{pmatrix}& -B_1 & -B_2 & \\-B_1 & & & -iE_1\\-B_2 & & & -iE_2\\& -iE_1 & -iE_2 & \end{pmatrix},$$ $$\delta g_{\mu\nu}^T=a^2\begin{pmatrix} & & & \\ & 2E^T_{11} & 2E^T_{12} & \\ & 2E^T_{12} & -2E^T_{11} & \\ & & & \end{pmatrix}=a^2\begin{pmatrix} & & & \\ & h_+ & h_\times & \\ & h_\times & -h_+ & \\ & & & \end{pmatrix}.$$

- $2E^T_{11}=h_+$ and $2E^T_{12}=h_\times$ are polarisation amplitudes of gravitational waves.
- We have also discussed about Gauge transformation in Fourier space.
- Several convention has been discussed e.g time derivative in Gauge transformation changed to conformal time derivative and spacial derivative $\xi^i_{,j}$ changed to $ik_j\xi^i$ in Fourier space.

## Summary (26/4/2022)-General Discussions on Codings
- Tried to run codes that saved in external harddisk but advised not to as it will be slower.
- Learnt to running codes using MobaXterm in UTAR RSC.
- Discussed on running Wolfram Language in Jupytar.

## Summary (9/5/2022)-Continuation of discussion on Cosmological Perturbation Theory
- We have discussed about the Fourier convention. Two methods have been discussed, the first method any vectors in space $\overrightarrow{v}(\overrightarrow{x})$ can be expressed as sum over a wavevector, $\overrightarrow{k}$ of corresponding vector in fourier space $\overrightarrow{v}_{\overrightarrow{k}}$ multiplied by exponential function, $e^{i\overrightarrow{k}.\overrightarrow{x}}$. Whereas in the second method, we tried to split any vectors in space into scalar and vector parts, then apply the divergence free condition to scalar and curl free condition to vectors. The first and second method is differ by phase.
- We have discussed the Scalar Perturbation and Conformal Newtonian Gauge.
- We also discussed the perturbations in curvature tensors, $R_{\alpha\beta}, R$ and Einstein tensor $G_{\alpha\beta}$.
## Summary (19/5/2022)
- We have discussed the perturbation in Energy tensor, $T^\mu_\nu=\bar{T^\mu_\nu}+\delta T^\mu_\nu$, where $\delta T^\mu_\nu$ is the perturbation.
- We have discused the perturbed four velocity, $\delta u$.
- We have discussed the separation of Energy tensor, $T^\mu_\nu$ into scalars, vectors and tensors.
## Summary (30/5/2022)
- We review back the Cosmological perturbation theory using softwares.
- We have compared our hand calculations with the one with using the software
## Summary - Physical discussion (15/6/2022)
- Review back the perturbation in Energy momentum tensors.
## Summary (23/6/2022)
- We proceed with understanding the misunderstanding in the Energy momentum tensor perturbation.
- Perturbation in Energy momementum tensor consist of two parts, one is the part that maintaining the perfect fluid form (5 degree of freedom) and the other is of non perfect fluid form (5 degree of freedom). Therefore one can treat the pertubation separately.
## Summary (30/6/2022)
- Review back our understanding of Perturbation in Energy momentum tensor.
- Discussed about the separation of Pertubation in Energy momentum tensors into scalars, vectors and tensors.
- Later, we try to compare the results with the Gauge transformation of Energy momentum tensor.
## Summary (6/7/2022)
- We have discussed the full perturbed Einstein's field equation.
- We discussed the Energy momentum continuity equation.
## Summary-Physical Discussion (13/7/2022)
- We discussed on deriving continuity equation for the perturbed Energy momentum tensor.
## Summary-Physical Discussion (27/7/2022)
- We diccused about field equations for perfect fluid scalar perturbations in Newtonian Gauge.
## Summary (17/8/2022)
- We discussed about perturbation in $f(R)$ gravity and reviewed field equation in perturbed $f(R)$ gravity from the paper "$f(R)$ gravity: scalar perturbations in the late Universe".
- We discussed about implementing perturbed $f(R)$ gravity in mathematica package xPand, which later can be extended to other modified gravity.
## Summary (24/8/2022)
- We discussed about cosmological perturbation theory in modified gravity e.g $f(R)$-gravity using mathematica.
- We tried to workout some calculations in cosmological perturbations in $f(R)$-gravity using mathematica.
## Summary (9/9/2022)
- We discussed about mathematica codings for computing cosmological perturbation in modified gravity e.g $f(R)$-gravity.
- We discussed about the energy conditions paper in modified gravity.
## Summary (23/9/2022)
- We have discussed about cosmological perturbation in $f(R)$ gravity using mathematica and the problems in the codings.
- We discussed about the energy condition paper and the way to plot the graph.
## Summary (29/9/2022)
- We have discussed the problems in coding for cosmological perturbation in $f(R)$-gravity and overcome some of the issues.
- We discussed the energy condition paper.
## Summary (6/10/2022)
- We discussed about energy condition paper, codings and plotting in Mathematica.
## Summary (14/10/2022)
- We discussed about the suitable plot for energy condition e.g contour plot or 3Dplot that can illustrated the values of the parameter clearly.
## Summary (21/10/2022)
- We discussed about finding the range for the EC to be satisfied and how mathematically find the ranges and ploting the graph too understand the ranges better.
## Summary (4/11/2022)
- We discussed about how to explain our results properly in a paper.
## Summary (10/11/2022)
- We discussed about the Energy condition in modified gravity.
- We amplified the errors in the computations for the $\sqrt(-Q)-$ modified gravity model.
- Several new analysis for the $Q^2$-model has been sugested e.g $p^m, p^{de}, \rho^{de},p^{eff},\rho^{eff}, \omega^{de}$ and $\omega^{eff}$.
# Summary (15/12/2022)
- We discussed about the Energy condition paper for modified gravity for $k=0$.
- Criticallly analysed the plots and discussions.
- Some comments on final modifications of the plot e.g size of the plots, structure of the paper and etc.
- Finally, discussed about sending the paper to the journal for publication after all the final corrections are made.
# Summary (21/12/2022)
- We discussed about the Energy condition paper for modified gravity for nonzero $k$.
- We disccussed about the cosmological parameters e.g q(t),j(t) to be used in the paper.
- The datas from observations only available for $k=0$.
- The work still under progress slowly.
# Summary (11/1/2023)
- We discussed about Cosmological Perturbation theory (cosmic time) in $f(R)-$gravity using computation software.
- Some of the perturbed $f(R)-$gravity equation is not well understood, therefore trying to get the perturbations in general relativity using computational software as the theory described in Weinberg's Cosmology book.
# Summary (19/1/2023)
- We discussed and revise the method of perturbation introduced in Weinberg's Cosmology book.
- In the Weinberg's Cosmology book, the author used trace of the energy-momentum tensor to replace the Ricci scalar. Modern book's did not use this method.
- We come to a conclussion to do perturbation in modern way in cosmic time without using the trace of energy-momentum tensor to replace the Ricci scalar for the case General relativity.
- Work under progress in perturbation: Do the perturbation in general relativity by hand (cosmic time), then proceed to perturbation in $f(R)-$gravity.
# Summary (2/2/2023)
- Discussed in detail about all the perturbed components in GR, e.g $\delta\Gamma^\alpha_{\beta\gamma},\delta R_{\alpha\beta},\delta R, \delta T_{\alpha\beta}$.
- Discussed the full perturbed field equations.
- Pending works: Perturbation in $f(R)-$gravity.
# Summary (10/2/2023)
- Discussed in detail about all the perturbed components in $f(R)-$ gravity, e.g $\delta(\nabla_\alpha\nabla_\beta f_R),\Box f_R$.
- Discussed each components of the modified perturbed Einstein tensors $\delta G^M_{\alpha\beta}$.
- Pending work: Complete the energy momentum parts and compare with other papers.
# Summary (24/3/2023)
- Discussed the perturbations formulas in $f(Q)-$gravity, e.g $\delta \Gamma^i_{\quad jk}, \delta \Gamma_{ijk}, \delta Q^i_{\quad jk},\delta Q_{ijk}, \delta P^i_{\quad jk}, \delta P_{ijk}$.
- Pending work: check the calculations e.g all the quantities as mentioned before and the field equation for $f(Q)-$gravity with some computation software.
# Summary (11/4/2023)
- Discussed about energy condition paper for correction needed before submitting to journal for publication and sent for preprint.
- Discussed about the $f(Q)-$gravity formulation for perturbations.
- Discussed about the density constrast equation for GR and posible extension to $f(Q)-$gravity.
# Summary (5/5/2023)
- Review about density contrast ($\delta$) equation and evolution of density contrast ($\dot \delta$) equation as discussed in last meeting.
- Discussed about CAMB implimentation in our $f(Q)-$gravity.
- Discussed about symbolic computation in CAMB and plotting for the resulting equations.
- Pending work:
1) What information we can extract and plot from the density contrast ($\delta$) and evolution of density contrast ($\dot \delta$).
2) Understanding the symbolic computation in CAMB and the respective plottings in detail.
# Summary (30/5/2023)
- Discussed about conservation of energy momentum tensor for perturbed spacetime
- Discussed about obtaining density contrast ($\delta$) and evolution equation ($\dot{\delta}$) by solving conservation equation for noninteracting and interactive fluids.
- Pending work:
1) $\delta\nabla T=\nabla\delta T?$ (done, it is same)
2) Derive conservation equations for multiple fluids and compare with other paper, how it is different for noninteracting and interacting fluids (in progress)
# Summary (15/6/2023)
- Discussed about the commutation of divergence of perturbation ($\nabla_\mu \delta T^\mu{}_\nu$) and perturbation of the divergence of a tensor ($\delta (\nabla_\mu T^\mu{}_\nu)$). Since perturbation is not a tensor, divergence cannot be applied to perturbation. Thus, $\delta\nabla T\neq\nabla\delta T$.
- Pending work:
1) Proving $\dot \delta$ obtained from field equation is same as the one obtained from energy momentum tensor. (proven by taking $f=Q(t)$)
2) Obtaining energy conservation equation using the method discussed in the paper "On the viablity of f(Q) gravity models by Avik de, Tee-How Loo".
# Summary (22/6/2023)
- Discussed about the computation of the obtained energy conservation equation base on the paper "On the viability of $f(Q)$ gravity model by Avik de, Tee-How Loo"
- Pending work:
1) Use mathematica to do the computations especially use some simple packages that can compute tensor computations without need to define anything.
2) Review "Cosmological perturbation in $f(T)$ gravity by Shih-Hung Chen, James B. Dent and et. al." paper to plot density contrast $\delta$ vs $log_{10}(1+z)$.
# Summary (8/7/2023)
- Reviewed the calculations in a particular section in the paper "Cosmological perturbation in $f(T)$ gravity by Shih-Hung Chen, James B. Dent and et. al.".
- Pending work:
1) Discuss about the detail calculations on how to find $\Phi$ for plotting the density contrast, $\delta$ vs $log_{10}(1+z)$.
# Summary (22/7/2023)
- Discussed the steps to arrive at a second order homogeneos differential equation with variable coefficient involving gravitional potential, $\Phi$ and solving this equation numerically in Python.
- Discussed about ploting the $\delta$ vs time, $t$ or scale factor, $a$ or resdhift, $z$ using the obtained $\Phi$ for matter dominated era
- Pending work
1) Read some papers on $f(Q)$ or $f(T)-$ gravity to refer the plots and to understand how they describe the plots in the paper.
2) The nature of sound speeds, $c_s{}^2$ and read/ note down some references.
3) Understand the $\Phi$ vs $a$ plots in Baumann book, try to plot the similar plot.
# Summary (5/10/2023)
- Discuss the derivation of $c_s{}^2=\frac{\partial p}{\partial \rho}\equiv\frac{\delta p}{\delta \rho}$ from fluid dynamic.
- Discuss the problems of solving the perturbation field equation in $f(Q,C)$ gravity.
# Summary (15//11/2023)
- Discuss about the codes for plotting of $\Phi$ vs $t$ in Python for $f(Q,C)-$ gravity.
# Summary (21/11/2023)
- Discuss about current progress in perturbation theory in $f(Q,C)-$ gravity.
- Discuss about the ploting of the results.
# Summary (5/12/2023)
- Discuss about Friedmann's like equations in $f(Q, C)-$ theory.
- Discuss about the $f(Q,C)-$ gravity in background spacetime e.g matter energy density, $\rho$, dark matter energy density, $\rho^{de}$, matter pressure, $p$ and dark matter pressure, $p^{de}$.
- Discuss about solving for Hubble parameter, $H(t)$ using matter pressure equation in $f(Q, C)-$ theory.
- Discuss about the initial condition that can be used to solve the $p=0$ equation numerically in $f(Q, C)-$ theory.
# Summary (28/12/2023)
- Discuss about the codes used to numerically solve for hubbles parameter, $H$, scale factor, $a$, gravitational potential, $\Phi$ and etc.
- Discuss about the units of hubble parameter, $H$ because of higher floating points in the numerical results.
- Discuss about NAN(not a number) in numerical results and replacing it to $0$.
- Pending work:
1) Check if by changing the units of H from km/s/Mpc to per bil yrs for the floating points to be reduced
2) Check the units of $\Phi$
# Summary (24/1/2024)
- Discuss the application of Quasi static limit in $f(Q, C)-$ gravity.
- It was mentioned in "Cosmology in $f(Q)-$ geometry" that in $f(Q)-$ theory, we can't apply strick QS approximation because the scalar sector becomes strongly coupled arround maximally symmetric background e.g. existences of de-Sitter solution in $f=Q+\beta Q^2$.
- In "Cosmological perturbation in $f(Q, T)-$ gravity", they ignored the time derivatives of the potentials e.g. $\dot \Phi, \dot \Psi$ and considered the sub-horizon or sub-hubble limit ($k>>aH$) to obtain an expression for $\Phi$ and $\Psi$ in term of $\delta\rho$ and $\bar \rho$.
- Pending work:
1) What it meant by scalar sector becomes strongly couple to the maximally symmetric background?
# Summary (14/2/2024)
- Discussed and reviewed the application of Quasi static approximation e.g ignoring the time derivatives of the potentials $\Phi, \Psi$.
- Applied the Quasi static and subhorizon approximation ($k>>aH$) in our perturbation equations for $f(Q, C)$, a detailed computatios have been reviewed.
- Comparing the results obtained for the density evolution equation $\ddot{\delta}$ to other existing modified gravity theories e.g.$f(T)-$, $f(Q)-$, $f(R)-$ theories. One can retrieve density evolution equation for $f(R)-$ gravity from the $f(Q, C)-$ gravity theory.

# Summary (18/3/2024)
- Discussed about solving for evolution of Hubble's parameter, H with scale factor, a using the matter density equation for the $f(Q,C)-$ gravity.
- Discussed about using the $H$ obtained previouslly to solve density evolution equation.
- Plotting $H$ vs $a$ for background $f(Q,C)-$ gravity and $\delta$ vs $a$.

# Summary (30/3/2024)
- Preparing and correcting a paper on Perturbation in $f(Q,C)-$ gravity.
- Reading and discussed about the computation of density growth, $f$ for $f(Q)-$ gravity.
- Discussed about obtaining an expression for $\alpha$ in $F=\alpha Q^n$, where $Q=6H^2$, in term of density parameter, $\Omega^{mo}$ and $E=\frac{H}{H_0}$.
- Discussed about converting time derivative in density evolution equation to derivatives w.r.t $ln(a)$.
# Summary (2/4/2024)
- Applying the steps learned before for $f(Q)-$ gravity to $f(Q,C)-$ gravity to derive density growth equations.
- Try to obtaining an expression for $\beta$ in $F=\beta C^n$, where $C=6(3H^2+\dot{H})$, in term of density parameter, $\Omega^{mo}$ and $E=\frac{H}{H_0}$.
# Summary (29/5/2024)
- Trying to understand cosmological perturbation for a most general perturbed metric $ds^2=-(1+2A)dt^2+2aB_idx^idt+a^2\left[(1+2D)\delta_{ij}+2E_{ij}\right]dx^idx^j$.
- Obtaining perturbation equations for the above mentioned metric.
# Summary (14/6/2024)
- Studying gauge transformation to understand Bardeen potential.
- Derivng all the equation to obtain bardeen potential for the metric $ds^2=-(1+2A)dt^2+2aB_idx^idt+a^2\left[(1+2D)\delta_{ij}+2E_{ij}\right]dx^idx^j$.
- Obtained Bardeen potentials are different from the one usually reported in the literature.
# Summary (28/6/2024)
- Derived perturbation equations for f(Q)- gravity for the metric $ds^2=-(1+2A)dt^2+2aB_idx^idt+a^2\left[(1+2D)\delta_{ij}+2E_{ij}\right]dx^idx^j$.
- The perturbation equations I obtained for the $f(Q)-$ gravity can be reduceable to GR by letting $f(Q)=Q$.
- Therefore, the results obtained for scalar perturbation in "Cosmology in $f(Q)$ cosmology" paper is wrong because it cannot be reduced to GR perturbation equation by setting $f(Q)=Q$. But they can be reduced to GR under Newtonian gauge ($B=0, E=0$).
# Summary (**/7/2024)
- Derived perturbation equations for $f(Q,C)-$ gravity for the metric $ds^2=-(1+2A)dt^2+2aB_idx^idt+a^2\left[(1+2D)\delta_{ij}+2E_{ij}\right]dx^idx^j$.