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Revisiting the Shell Crossing Singularity inside LTB model

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### Revisiting the Shell Crossing Singularity inside LTB model

Antonio Enea Romano

Hsu-Wen Chiang

LeCosPA, NTU

Future Study

- Instability of LTB model

Krzysztof Bolejko, Charles Hellaby

- Instability Analysis

Cooperation

- Analysis of LTB model in low redshift region

- Initial Condition Relationship

- Numerical analysis of LTB model

Antonio Enea Romano

Antonio Enea Romano

- Solution to high redshift

- Choose M proportional to r3 and tB=0

- Evolution Equations of LTB model

Cooperation

- Simulation of realistic initial condition

Cooperation

- Done in last two weeks

Antonio Enea Romano

Antonio Enea Romano

Cooperation

- Search most stable initial condition

- Done before

- Comparing with CMB

- Comparing with BAO data

- initial condition for CMB data

- Doing or will be done

Adam Moss, James P. Zibin, Douglas Scott

- Solution for CMB data

Cooperation

Cooperation

Hsu-Wen Chiang

Cooperation

1

From “Do recent accurate measurements of H0 really rule out void models as alternatives to dark energy?” --Antonio Enea Romano --10, May, 2011

What is done beforeVoid Model

ΛCDM Model

Pro: Explain most of data with two variables, simple

Con: Origin of cosmological constant remains unknown

- Pro: No new physics introduced
- Con: Too many possibilities, violate Copernican principle

Expressed by LTB metric, which is a generalization of FRW metric to inhomogeneous universe.

M in FRW is the same as mass, so for simplicity and compatibility we should set it proportional to r3, to wit ρ0 is constant (FLRW gauge)

Variable Transformation to FRW-like form

E acts like the spatial curvature

Change to conformal timeη=∫dt/a(t,r)+tb(r) , the integration constant tb is called bang function

From now on we set tb=0

2

Some part is from “Do recent accurate measurements of H0 really rule out void models as alternatives to dark energy?” --Antonio Enea Romano --10, May, 2011

What is done before- After some lengthy calculation we’ll get a set of underdetermined first order differential equations describing the evolution of curvature parameter k, radius r, and the conformal time η (η=t/a) along radical geodesic in unit of redshift z.
- To make this “evolution equations” unique we need to introduce the

confinement of observation data from luminosity distance.

- But before doing numerical calculation of the evolution equations, we first need to set initial conditions correctly, which is the main topic of the report in the last meeting.

After some calculation, we get

(in unit of H0 and a0)

This result shows that the LTB model compatible with ΛCDM model is fully determined by single parameter K0(spatial curvature at the center), which is not known before.

3

Some part is from “Some Properties Of Singularities In The Tolman Model” --Charles Hellaby --1985

Instability Analysis of LTB model- It is known that there are 3 kinds of singularities existing in LTB model. One lays on past infinity and future infinity, also known as big bang and big crunch. One happens when the shell of constant time starts to contract/expand , called shell-crossing singularity(SCS), and is the main source of instability. Last one is named as ESC singularity, which does not appear if the initial condition is chose as what we mentioned in page3.
- Now the only singularity needed to take care of is SCS. The characteristic of SCS is shown below.

In addition kr2 reaches its extrema at SCS as well

Energy density

∂zR

Radius of the shell

-(spatial curvature)

∂rR

SCS(∂rR=∂zR=0)

But for our model since the effect of SCS happens earlier than the event horizon, we should stop integration before the energy density starts to fall.

EH of SCS

K0=-0.95

Ωm=0.738

Point where LTB starts to fail (local extrema of energy density)

5

Instability Analysis of LTB model

- Although it’s best to stop integration and jump to other methods before the local maximum of energy density, we can’t know the exact position of local maximum before integration.
- Thus we decide to use ∂zR=0 as reference point since it’s the same point as the maximum of angular luminosity distance (Da=R). The corresponding z we call it “zmax”.

Stops at zmax-0.1(z~1.555)

Stops after 1M steps(z~2.025)

K0=-0.95

Ωm=0.738

∂zR=0

SCS

EH of SCS

Energy density

∂zR

Radius of the shell

-(spatial curvature)

∂rR

Stopping point

6

Most Stable Solution

- Let’s look at the behavior of SCS in the “most stable solution”. It’s clear that the event horizon in the original SCS disappears. There’re 2 possible explanation upon this phenomenon.
- First, this degenerated SCS is resulted from numerical problem(0/0). From purely analytical study it’s shown that such kind of degeneration is from the fact that this is a coordinate singularity called regular extrema. Unlike SCS, regular extrema has finite energy density.

But after calculating numerator of energy density(∂rR is denominator of energy density) we find that it doesn’t goes to zero, so this guess work is ruled out.

- Second, it’s possible that this degenerated SCS is a naked singularity just like ESC singularity. This assumption is supported by the behavior of ∂rR in the numerical simulation below.
- To answer this question, further study is needed.

Until now, it remains open.

Energy density

∂zR

Radius of the shell

-(spatial curvature)

∂rR

8

From “Role of shell crossing on the existence and stability of trapped matter shells in spherical inhomogeneous ΛCDM models”-- Morgan Le Delliou, Filipe C. Mena, and Jose´ P. Mimoso --1985

Shell Crossing Singularity in LTB model- Since in LTB model E represents the local spatial curvature, the trajectory of shell dM corresponding to that radius then can be determined by its initial velocity and local E.

From “Role of shell crossing on the existence and stability of trapped matter shells in spherical inhomogeneous ΛCDM models”-- Morgan Le Delliou, Filipe C. Mena, and Jose´ P. Mimoso --1985

Shell Crossing Singularity in LTB model
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