Bianchi Cosmologies with Anisotropic Matter: Locally Rotationally Symmetric Models

TL;DR

This paper explores the dynamics of Bianchi cosmological models with anisotropic matter, revealing conditions for non-recollapsing solutions and oscillatory behavior near singularities, thus highlighting the significant impact of matter anisotropy.

Contribution

It provides a comprehensive analysis of Bianchi types I, II, and IX with anisotropic matter, extending understanding beyond perfect fluids and vacuum models.

Findings

Existence of non-recollapsing solutions in Bianchi type IX with anisotropic matter.

Vlasov matter models exhibit oscillatory behavior near singularities.

Anisotropic matter significantly alters cosmological dynamics compared to perfect fluids.

Abstract

The dynamics of cosmological models with isotropic matter sources (perfect fluids) is extensively studied in the literature; in comparison, the dynamics of cosmological models with anisotropic matter sources is not. In this paper we consider spatially homogeneous locally rotationally symmetric solutions of the Einstein equations with a large class of anisotropic matter models including collisionless matter (Vlasov), elastic matter, and magnetic fields. The dynamics of models of Bianchi types I, II, and IX are completely described; the two most striking results are the following: (i) There exist matter models, compatible with the standard energy conditions, such that solutions of Bianchi type IX (closed cosmologies) need not necessarily recollapse; there is an open set of forever expanding solutions. (ii) Generic type IX solutions associated with a matter model like Vlasov matter exhibit oscillatory behavior toward the initial singularity. This behavior differs significantly from that of vacuum/perfect fluid cosmologies; hence "matter matters". Finally, we indicate that our methods can probably be extended to treat a number of open problems, in particular, the dynamics of Bianchi type VIII and Kantowski-Sachs solutions.

Figures

Tables

Table 1. Table 1: This table gives the classification of Bianchi class A models into the different Bianchi types . Each Bianchi type is defined by the values of the triple ( n ^ 1 , n ^ 2 , n ^ 3 ) subscript ^ 𝑛 1 subscript ^ 𝑛 2 subscript ^ 𝑛 3 (\hat{n}_{1},\hat{n}_{2},\hat{n}_{3}) ; this is up to permutations and overall changes of sign of the constants (which correspond to discrete symmetries of the frame).

Bianchi	${\hat{n}}_1$	${\hat{n}}_2$	${\hat{n}}_3$
type
I	0	0	0
II	1	0	0
VI0	0	1	$-1$
VII0	0	1	1
VIII	$-1$	1	1
IX	1	1	1

Table 2. Table 2: Anisotropy classification of matter sources according to the value of the anisotropy parameter β 𝛽 \beta . The value β = 0 𝛽 0 \beta=0 is exceptional—it corresponds to a matter source which behaves like a perfect fluid ‘at the singularity’. In the second column we give the range of β 𝛽 \beta and w 𝑤 w that guarantees the validity of the strong and dominant energy condition, see  [ 5 ] for more details.

Type	$\beta$ values	Energy conditions require
D-	$\beta \le -2$	$\beta =-2,w\ge \frac{1}{3}$
C-	$\beta \in (-2,-1)$	$w>0,\beta \ge \max \{-2,-\frac{1+w}{1-w}\}$
B-	$\beta =-1$	$w\ge 0$
A-	$\beta \in (-1,0)$	$w\ge -\frac{1}{3},\beta \ge \max \{-1,-\frac{1+w}{1-w}\}$
​A 0	$\beta =0$	$w\ge -1/3$
A+	$\beta \in (0,1)$	$w\ge -1/3$
B+	$\beta =1$	$w\ge -1/3$
C+	$\beta \in (1,2)$	violated
D+	$\beta \ge 2$	violated

Table 3. Table 3: The past and the future attractor for Bianchi type I orbits encode the asymptotic behavior of typical solutions. The possible α 𝛼 \alpha / ω 𝜔 \omega -limits of non-generic orbits can be read off from Fig.  2 .

Type	Past Attractor	Future Attractor
D-	$T_{♭}$, $Q_{♯}$	$T_{♯}$, $Q_{♭}$
C-, B-	$T_{♭}$, $Q_{♯}$	$R_{♯}$, $Q_{♭}$
A-, A0-	$T_{♭}$, $Q_{♯}$	$R_{♯}$, $R_{♭}$
A+, A0+	$T_{♭}$, $Q_{♯}$	$F$
B+, C+	$Q_{♯}$	$F$
D+	Heteroclinic cycle (47)	$F$

Table 4. Table 4: Past and future attractor of orbits in 𝒳 II subscript 𝒳 II \mathcal{X}_{\,\mathrm{II}} , which represent LRS Bianchi type II cosmological models. The exact solutions represented by the fixed points are given in Sec.  5 and in Appendix  A .

Type	Past Attractor	Future Attractor
B-, C-, D-	$T_{♭}$	$Q_{♭}$
A-($\beta \le {\beta }_{♭}$)	$T_{♭}$	$R_{♭}$
A-(), A0, A+	$T_{♭}$	$C_{♭}$
B+, C+	Heteroclinic cycle	$C_{♭}$
D+	Heteroclinic cycle/network	$C_{♭}$

Table 5. Table 5: Possible α 𝛼 \alpha - and ω 𝜔 \omega -limit sets of LRS Bianchi type II cosmological models. Each Bianchi type II solution corresponds to an orbit in 𝒳 II subscript 𝒳 II \mathcal{X}_{\,\mathrm{II}} ; hence the intersection of the unstable [stable] manifold of a fixed point with 𝒳 II subscript 𝒳 II \mathcal{X}_{\,\mathrm{II}} yields the set of type II solutions converging to that fixed point as t → 0 → 𝑡 0 t\rightarrow 0 [ t → ∞ → 𝑡 t\rightarrow\infty ]. If this set is three-dimensional, the fixed point is a source [sink]; if the dimension is two, there exists a one-parameter set of orbits converging to the fixed point; if the dimension is one, there is only one orbit with that property. Center manifold analysis shows that the cases A 0+  and A 0-  behave like A +  ( 0 < β < β ♯ 0 𝛽 subscript 𝛽 ♯ 0<\beta<\beta_{\sharp} ) and A -  ( β ♭ < β < 0 subscript 𝛽 ♭ 𝛽 0 \beta_{\flat}<\beta<0 ), respectively.

Fixed point	Dimension of unstable [stable] manifold intersected with ${𝒳}_{\mathrm{II}}$
	D-, C-, B-	A-($\beta \le {\beta }_{♭}$)	A-()	A+()	A+(${\beta }_{♯}\le \beta$)	B+, C+, D+
$T_{♭}$	3	3	3	3	3
$Q_{♭}$	[3]
$R_{♭}$		[3]		2	2
$R_{♯}$				2
$F$	2	2	2	1	1	1
$C_{♭}$			[3]	[3]	[3]	[3]
$C_{♯}$	1	1	1	1
het. cycle						3

Table 6. Table 6: Possible α 𝛼 \alpha -limit sets of LRS Bianchi type IX models and ω 𝜔 \omega -limit sets of forever expanding LRS Bianchi type IX models. Each orbit in 𝒴 IX + superscript subscript 𝒴 IX \mathcal{Y}_{\mathrm{IX}}^{+} corresponds to (the expanding phase of) a Bianchi type IX solution (and conversely); hence the intersection of the unstable [stable] manifold of a fixed point with 𝒴 IX + superscript subscript 𝒴 IX \mathcal{Y}_{\mathrm{IX}}^{+} yields the set of type IX solutions converging to the solution represented by that fixed point as t → 0 → 𝑡 0 t\rightarrow 0 [ t → ∞ → 𝑡 t\rightarrow\infty ]. If this set is three-dimensional, the fixed point is a source [sink]; if the dimension is two, there exists a one-parameter set of orbits converging to the fixed point; if the dimension is one, there is only one orbit with that property. Center manifold analysis shows that the cases A 0+  and A 0-  behave like A +  ( 0 < β < β ♯ 0 𝛽 subscript 𝛽 ♯ 0<\beta<\beta_{\sharp} ) and A -  (not A - IX ), respectively. The exact solutions represented by the fixed points are given in Sec.  5 and in Appendix  A .

Fixed point	Dimension of unstable [stable] manifold intersected with ${𝒴}_{\mathrm{IX}}^{+}$
	D-	C-	B-	A- (not A-IX)	A-IX	A+()	A+(${\beta }_{♯}\le \beta$)	B+	C+	D+
$T_{♭}$	3	3	3	3	3	3	3
$R_{♭}$					[3]	2	2
$R_{♯}$						2
$F$	2	2	2	2	2	1	1	1	1	1
$C_{♯}$	1	1	1	1	1	1
het. cycle								3	3	3
$P$					[2]

Table 7. Table 7: Table of the fixed points of the reduced dynamical system. Each of the points R ♭ subscript R ♭ \mathrm{R}_{\flat} , R ♯ subscript R ♯ \mathrm{R}_{\sharp} , C ♭ subscript C ♭ \mathrm{C}_{\flat} , C ♯ subscript C ♯ \mathrm{C}_{\sharp} , and P P \mathrm{P} exists only under certain restrictions on the anisotropy parameter β 𝛽 \beta .

Fixed	$s$	${\Sigma }_{+}$	$M_1^2$	$H_D$	Solution	Bianchi
point						type
$T_{♭}$	$0$	$-1$	$0$	$1$	(30)	I
$T_{♯}$	$\frac{1}{2}$	$-1$	$0$	$1$	(30)	I
$Q_{♭}$	$0$	$1$	$0$	$1$	(31)	I
$Q_{♯}$	$\frac{1}{2}$	$1$	$0$	$1$	(31)	I
$F$	$\overline{s}$	$0$	$0$	$1$	(32)	I
$R_{♭}$	$0$	$-\beta$	$0$	$1$	(101)	I
$R_{♯}$	$\frac{1}{2}$	$\frac{\beta }{2}$	$0$	$1$	(100)	I
$C_{♭}$	$0$	$\frac{1+3w}{8+3\beta \left(1-w\right)}$	$\frac{36\left(1-w\right)\left[3{\beta }^2\left(1-w\right)+8\beta +\left(1+3w\right)\right]}{{\left[8+3\beta \left(1-w\right)\right]}^2}$	$1$	(103)	II
$C_{♯}$	$\frac{1}{2}$	$\frac{1+3w}{16-3\beta \left(1-w\right)}$	$\frac{36\left(1-w\right)\left[3{\beta }^2\left(1-w\right)-16\beta +4\left(1+3w\right)\right]}{{\left[16-3\beta \left(1-w\right)\right]}^2}$	$1$	(102)	II
$P$	$0$	$\frac{1+3w}{\sqrt{{\left(1-3w\right)}^2+6\beta \left(1-w\right)}}$	$0$	$\frac{2+3\beta \left(1-w\right)}{\sqrt{{\left(1-3w\right)}^2+6\beta \left(1-w\right)}}$	(104)	III/KS

Equations

$${}^4\mathbf{g}=-d{t}^2+g_{ij}(t){\omega }^i{\omega }^j.$$

$$R-k^{ij}{k}_{ij}+(g^{ij}{k}_{ij}{)}^2=2T_{00},{\nabla }_i{k}_j^i-{\nabla }_j(g^{lm}{k}_{lm})=-T_{0j},$$

$${\partial }_t{g}_{ij}=-2k_{ij},{\partial }_t{k}_j^i=R_j^i+\left(g^{lm}{k}_{lm}\right)k_j^i-T_j^i+\frac{1}{2}{\delta }_j^i(g^{lm}{T}_{lm}-T_{00}).$$

$$R_{ij}=-\frac{1}{2}{C}_{ki}^l\left(C_{lj}^k+g_{lm}{g}^{kn}{C}_{nj}^m\right)-\frac{1}{4}{C}_{nk}^m{C}_{ql}^p{g}_{jm}{g}_{ip}{g}^{kq}{g}^{ln}.$$

$$C_{ij}^k={\epsilon }_{ijl}{n}^{kl},$$

$${\epsilon }_{kij}{n}^{kl}{k}_l^i=-T_{0j}.$$

$${\hat{n}}^{ij}=\mathrm{diag}({\hat{n}}_1,{\hat{n}}_2,{\hat{n}}_3).$$

$$d{\hat{\omega }}^1=-{\hat{n}}_1{\hat{\omega }}^2\wedge {\hat{\omega }}^3,d{\hat{\omega }}^2=-{\hat{n}}_2{\hat{\omega }}^3\wedge {\hat{\omega }}^1,d{\hat{\omega }}^3=-{\hat{n}}_3{\hat{\omega }}^1\wedge {\hat{\omega }}^2.$$

$${\partial }_{t}R=2R^{ij}{k}_{ij}.$$

$${\partial }_{t}R=2k^{ij}{R}_{ij}+g^{ij}{\partial }_t{R}_{ij},$$

$$g^{ij}{\partial }_t{R}_{ij}$$

$$\displaystyle\quad-{\textstyle\frac{1}{2}}\,g^{ij}\,\hat{C}^{m}_{\ nk}\hat{C}^{p}_{\ ql}\,\big{(}-k_{jm}g_{ip}g^{kq}g^{ln}-k_{ip}g_{jm}g^{kq}g^{ln}+k^{kq}g_{jm}g_{ip}g^{ln}+k^{ln}g_{jm}g_{ip}g^{kq}\big{)}$$

$$={\hat{C}}_{ki}^l{\hat{C}}_{nj}^m{k}_{lm}{g}^{ij}{g}^{kn}-{\hat{C}}_{ki}^l{\hat{C}}_{nj}^m{k}^{kn}{g}^{ij}{g}_{lm}+{\hat{C}}_{nk}^m{\hat{C}}_{ql}^p{k}_{pm}{g}^{kq}{g}^{ln}-{\hat{C}}_{nk}^m{\hat{C}}_{ql}^p{k}^{kq}{g}_{mp}{g}^{ln}$$

$$=0,$$

$$T_{\mu\nu}=\rho\,(dt\otimes dt)_{\mu\nu}+p\big{[}{}^{4}\mathbf{g}+dt\otimes dt\big{]}_{\mu\nu}\>,$$

$$p=w\rho ,w=\mathrm{const}.$$

$$\rho ={\rho }_0{n}^{1+w},{\rho }_0=\mathrm{constant},n=(\det g{)}^{-1/2}.$$

$$T_{00}=\rho =\rho (g_{ij})$$

$$T_j^i=-2\frac{\partial \rho }{\partial g_{il}}{g}_{jl}-{\delta }_j^i\rho .$$

$${\partial }_t\rho =k_i^j(T_j^i+{\delta }_j^i\rho ).$$

$${\partial }_t\rho =-2k_i^j{g}_{jl}\frac{\partial \rho }{\partial g_{jl}}.$$

$$m_i:=\sqrt{\frac{g_{ii}}{g_{jj}{g}_{kk}}}=\sqrt{\frac{g^{jj}{g}^{kk}}{g^{ii}}}.$$

$$R_i^i=\frac{1}{2}\left[{\hat{n}}_i^2{m}_i^2-{\left({\hat{n}}_j{m}_j-{\hat{n}}_k{m}_k\right)}^2\right].$$

R=n^i​mi​(n^j​mj​+n^k​mk​−21​n^i​mi​)−21​(n^j​mj​−