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Negative spatial curvature: hyperbolic models 

In hyperbolic models or regions containing a center, Γ > 0 and δ ( k ) i ≥ − 2 / 3 hold everywhere (conditions (61) and (68)), therefore, as a consequence of (37) and (B1), we have: 

Hence, since all regular hyperbolic models or regions comply with k$_{q}$ < 0, then all these models or regions have also negative local spatial curvature. As explain further below, the converse is not true, as local spatial curvature can be negative in certain elliptic regions in which k$_{q}$ ≥ 0. 

Elliptic models and positive spatial curvature. 

The relation between k$_{q}$ and k is more complicated in elliptic models or regions, since standard regularity (conditions (61) and (72)) do not place a lower bound on δ ( k ) i . Hence, the possibility that δ ( k ) i < − 2 / 3 occurs cannot be ruled out, and so k < 0 can happen in regions where k$_{q}$ > 0. It is straightforward to show from (62) that δ ( k ) i ≤ − 2 / 3 can only occur in a regular elliptic model or region (Γ > 0) if E ′ = 0 for some 0 < r$_{tv}$ < r in a domain ϑ ( r ). As a consequence, we have δ ( k ) i > − 2 / 3 for all regular elliptic models in which E ′ ≤ 0 holds for all r (with E ′ = 0 only at the symmetry center), and so (B1) 

[1] Lemaˆıtre G 1933 Ann. Soc. Sci. Brux. A 53 51. See reprint in Lemaˆıtre G 1997 Gen. Rel. Grav. 29 5; Tol- man R C 1934 Proc. Natl Acad. Sci. 20 169; Bondi H 1947 Mon. Not. R. Astron. Soc. 107 410. 

[2] Krasi´ nski A, Inhomogeneous Cosmological Models , Cam- bridge University Press, 1998. 

[3] Plebanski J and Krasinski A, An Introduction to General Relativity and Cosmology , Cambridge University Press, 2006. 

[4] Eardley D M 1974 Commun Math Phys 37 287; Eardley D M and Smarr L 1979 Phys Rev D 19 2239; Dyer C C 1979 MNRAS 189 189; Waugh B and Lake K 1988 Phys Rev D 38 1315; Waugh B and Lake K 1989 Phys Rev D 40 2137; Lemos J P S 1991 Phys Lett A 158 279 

[5] Joshi P S and Dwivedi I H 1993 Phys Rev D 47 5357; Joshi P S and Singh T P 1995 Phys Rev D 51 6778; Dwivedi I H and Joshi P S 1997 Class. Quant. Grav. 47 5357 

[6] Vaz C, Witten L and Singh T P 2001 Phys Rev D 63 104020; Kiefer C, Mueller-Hill, Vaz C 2006 Phys Rev D 73 044025; Bojowald M, Harada T and Tibrewala R 2008 Phys Rev D 78 064057 

[7] Krasi´ nski A and Hellaby C 2002 Phys Rev D 65 023501 

[8] Krasi´ nski A and Hellaby C 2004 Phys Rev D 69 023502 

[9] Krasi´ nski A and Hellaby C 2004 Phys Rev D 69 043502 

[10] Hellaby C and Krasi´ nski A 2006 Phys Rev D 73 023518 

[11] Pascual–S´anchez J F 1999 Mod. Phys. Lett. A 14 1539; Sugiura N K and Harada T 1999 Phys Rev D 60 103508; Celeri`er M N 2000 Astron. Astrophys. 353 63; Tomita K 2001 MNRAS 326 287; Iguchi H, Nakamura T and 

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implies in this case that k$_{q}$ > 0 ⇒ k > 0 everywhere. If there is a zero of E ′ at some r = r $^{∗}$, then δ ( k ) < 2 

i − / 3 will hold in some regions without violating regularity con- ditions. Since − 1 ≤ E ≤ 0 and E (0) = 0, then for r ≈ 0 we must have E ′ < 0. Thus, the only possible configu- ration is: E ′ ≤ 0 and δ ( k ) i > − 2 / 3 for 0 ≤ r < r $^{∗}$, with E ′ ≥ 0 and δ ( k ) i < − 2 / 3 for r > r $^{∗}$. Further insight into this situation comes from rewriting (13) as 

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If the regularity condition (A7) holds, then R ′ $_{/}$√$_{1 +}$$_{E >}$ 0, and so for E $^{′}$( r $^{∗}$) = 0 to occur the local curvature k must decrease sufficiently to reach k ( r $^{∗}$) = k$_{q}$ ( r $^{∗}$) / 3, hence E ′ > 0 (or δ ( k ) i < − 2 / 3) leads to further decreas- ing of k , so that k ( r $^{∗}$) < k$_{q}$ ( r $^{∗}$) / 3 holds for r > r $^{∗}$. A situation in which k < 0 occurs with k$_{q}$ > 0 can easily be conceived: since k ′ and k ′ q are both monotonously nega- tive, we have a curvature clump and so k$_{q}$ > k , thus, if k$_{q}$ decays to zero sufficiently fast k might become neg- ative. This happens in the elliptic side of the mixed el- liptic/hyperbolic configuration examined in section XIV D. 

Nakao K 2002 Prog. Theor. Phys. 108 809; Schwarz D J 2002 Accelerated expansion without dark energy Preprint arXiv:astro-ph/0209584v2 ; 

[12] Apostolopoulos P et al 2006 JCAP P06 009; Kai T, Kozaki H, Nakao K, Nambu Y and Yoo C M 2007 Prog. Theor. Phys. 117 229-240 ( Preprint arXiv:gr-qc/0605120 ); Mattsson T and Ronkainen M 2008 JCAP 0802 004 ( Preprint arXiv:astro-ph/0708.3673v2 ); Bolejko K and Anders- son L 2008 JCAP 10 003 ( Preprint arXiv:0807.3577 ) 

[13] Kolb E W, Matarrese S, Notari A and Riotto A 2005 Phys Rev D 71 023524 ( Preprint arXiv:hep-ph/0409038v2 ); Marra V, Kolb E W and Matarrese S 2008 Phys Rev D 77 023003; Marra V, Kolb E W, Matarrese S and Riotto A 2007 Phys Rev D 76 123004. 

[14] Chuang C H, Gu J A and Hwang W Y P 2005 Class.Quant.Grav. , 25 , 175001 Preprint astro-ph/0512651 

[15] Rasanen S 2006 Class. Quant. Grav. 23 1823-1835; En- qvist K and Mattsson T 2007 JCAP 0702 019 ( Preprint arXiv:astro-ph/0609120v4 ); Enqvist K 2008 Gen. Rel. Grav. 40 451-466 (Preprint arXiv:0709.2044 ) 

[16] Garc´ıa–Bellido J and Troels H 2008 JCAP 0804:003 Preprint gr-qc/0802.1523v3 [astro-ph] 

[17] Moffat J W 2006 J. Cosmol. Astropart. Phys. JCAP(2006)001; Alnes H, Amazguioui M and Gron O 2006 Phys Rev D 73 083519; Alnes H and Amazguioui M 2006 Phys Rev D 74 103520; Alnes H and Amazguioui M 2006 Phys Rev D 75 023506; 

[18] Bolejko K, Krasi´ nski A and Hellaby C 2005 MNRAS 362 

