Cosmological Perturbation Theory in the Presence of Non-Linear Structures

Transcription

1 Cosmological Perturbation Theory in the Presence of Non-Linear Structures Timothy Clifton Queen Mary University of London, UK GR Effects in Cosmological Large-Scale Structure Meeting, Sexten Center for Astrophysics 16 th -20 th July 2018

2 Gravity as a weak field

3 Gravity as a weak field In the absence of non-linear structure: Can take Apply perturbation theory (i.e. expand all equations perturbatively, and solve order by order)

4 Gravity as a weak field In the absence of non-linear structure: Can take Apply perturbation theory (i.e. expand all equations perturbatively, and solve order by order) requires v i ~δ~φ etc.

5 Gravity as a weak field In the absence of non-linear structure: Can take Apply perturbation theory (i.e. expand all equations perturbatively, and solve order by order) requires v i ~δ~φ etc. In the presence of non-linear structure: Treat gravity as Newtonian, with Use the non-linear Eulerian equations of hydrodynamics

6 Gravity as a weak field In the absence of non-linear structure: Can take Apply perturbation theory (i.e. expand all equations perturbatively, and solve order by order) requires v i ~δ~φ etc. In the presence of non-linear structure: Treat gravity as Newtonian, with Use the non-linear Eulerian equations of hydrodynamics the leading-order part of a post-newtonian expansion

7 Gravity as a weak field In the absence of non-linear structure: Can take Apply perturbation theory (i.e. expand all equations perturbatively, and solve order by order) requires v i ~δ~φ etc. In the presence of non-linear structure: Treat gravity as Newtonian, with Use the non-linear Eulerian equations of hydrodynamics the leading-order part of a post-newtonian expansion not necessarily convergent when treated as perturbation theory

8 Newtonian gravity vs perturbation theory This difference between these expansions is important: Linear theory One loop correction Two loop correction [Carlson et al, Phys. Rev. D80, (2009)]

9 Newtonian gravity vs perturbation theory This difference between these expansions is important: Linear theory One loop correction Two loop correction does not converge in the highly non-linear regime [Carlson et al, Phys. Rev. D80, (2009)]

10 Gravity on small scales

11 Gravity on small scales Einstein s equations can be thought of as a wave equation will null characteristics. Solutions look like: (t,x) C

12 Gravity on small scales Einstein s equations can be thought of as a wave equation will null characteristics. Solutions look like: (t,x) C Consider a non-linear system: L N

13 Gravity on small scales Einstein s equations can be thought of as a wave equation will null characteristics. Solutions look like: (t,x) C Consider a non-linear system: L N

14 Gravity on small scales Einstein s equations can be thought of as a wave equation will null characteristics. Solutions look like: (t,x) C Consider a non-linear system: L N characteristic time scale of system

15 Gravity on small scales Einstein s equations can be thought of as a wave equation will null characteristics. Solutions look like: (t,x) C Consider a non-linear system: L N characteristic time scale of system λ c

16 Gravity on small scales Einstein s equations can be thought of as a wave equation will null characteristics. Solutions look like: (t,x) C Consider a non-linear system: characteristic length scale of gravitational field L N characteristic time scale of system λ c

17 Gravity on small scales Einstein s equations can be thought of as a wave equation will null characteristics. Solutions look like: (t,x) C Consider a non-linear system: characteristic length scale of gravitational field L N characteristic time scale of system λ c

18 Gravity on small scales Einstein s equations can be thought of as a wave equation will null characteristics. Solutions look like: (t,x) C Consider a non-linear system: characteristic length scale of gravitational field L N characteristic time scale of system λ c

19 Gravity on small scales Einstein s equations can be thought of as a wave equation will null characteristics. Solutions look like: (t,x) C Consider a non-linear system:

20 Gravity on small scales Einstein s equations can be thought of as a wave equation will null characteristics. Solutions look like: (t,x) C Consider a non-linear system:

21 Gravity on small scales Einstein s equations can be thought of as a wave equation will null characteristics. Solutions look like: (t,x) C Consider a non-linear system:

22 Gravity on small scales Einstein s equations can be thought of as a wave equation will null characteristics. Solutions look like: (t,x) C Consider a non-linear system: Changed the characteristic of the equation

23 Gravity on small scales Einstein s equations can be thought of as a wave equation will null characteristics. Solutions look like: (t,x) C Consider a non-linear system: only a good approximation on small scales Changed the characteristic of the equation

24 Gravity on small scales Einstein s equations can be thought of as a wave equation will null characteristics. Solutions look like: (t,x) C Consider a non-linear system:

25 Gravity on small scales Einstein s equations can be thought of as a wave equation will null characteristics. Solutions look like: (t,x) C Consider a non-linear system: and

26 Gravity on small scales Einstein s equations can be thought of as a wave equation will null characteristics. Solutions look like: (t,x) C Consider a non-linear system: and

27 Gravity on small scales Einstein s equations can be thought of as a wave equation will null characteristics. Solutions look like: (t,x) C Consider a non-linear system: and for all matter and gravitational fields

28 Gravity on small scales Einstein s equations can be thought of as a wave equation will null characteristics. Solutions look like: (t,x) C Consider a non-linear system: and required from dimensionality for all matter and gravitational fields

29 Gravity on small scales Einstein s equations can be thought of as a wave equation will null characteristics. Solutions look like: (t,x) C Consider a non-linear system: and required from dimensionality has not assumed anything about the size of density contrasts for all matter and gravitational fields

30 Gravity on small scales Einstein s equations can be thought of as a wave equation will null characteristics. Solutions look like: (t,x) C Consider a non-linear system: and required from dimensionality has not assumed anything about the size of density contrasts for all matter and gravitational fields Can continue to apply to higher and higher orders, to determine relativistic effects in the presence of non-linear structures [see e.g. Poisson & Will, Gravity]

31 Building a cosmology with non-linear structures

32 Building a cosmology with non-linear structures Consider a region of space that is small enough that it can be considered as perturbed Minkowski space: ~ 100 Mpc

33 Building a cosmology with non-linear structures Consider a region of space that is small enough that it can be considered as perturbed Minkowski space: ~v L C ~ 100 Mpc

34 Building a cosmology with non-linear structures Consider a region of space that is small enough that it can be considered as perturbed Minkowski space: ~v L C ~ 100 Mpc

35 Building a cosmology with non-linear structures Consider a region of space that is small enough that it can be considered as perturbed Minkowski space: ~v L C ~ 100 Mpc

36 Building a cosmology with non-linear structures Consider a region of space that is small enough that it can be considered as perturbed Minkowski space: apply junction conditions at the boundary ~v L C ~ 100 Mpc

37 Building a cosmology with non-linear structures Consider a region of space that is small enough that it can be considered as perturbed Minkowski space: apply junction conditions at the boundary ~v L C ~ 100 Mpc - Junction conditions provide boundary conditions for solving PDEs inside. - A global space-time emerges, isometric to a perturbed RW geometry. - The large-scale expansion is given by:

38 [Sanghai & TC, PRD 91, (2015); PRD 94, (2016)] Building a cosmology with non-linear structures Consider a region of space that is small enough that it can be considered as perturbed Minkowski space: apply junction conditions at the boundary ~v L C ~ 100 Mpc - Junction conditions provide boundary conditions for solving PDEs inside. - A global space-time emerges, isometric to a perturbed RW geometry. - The large-scale expansion is given by: effective fluid

39 Building a cosmology with non-linear structures This approach provides: - explicit links between the (post-newtonian) gravitational fields of nonlinear structures on small scales, and the large-scale expansion of space. - a way to relate perturbations to a cosmological RW space, and post- Newtonian perturbations to Minkowski space.

40 Building a cosmology with non-linear structures This approach provides: - explicit links between the (post-newtonian) gravitational fields of nonlinear structures on small scales, and the large-scale expansion of space. - a way to relate perturbations to a cosmological RW space, and post- Newtonian perturbations to Minkowski space. dark energy parameters and PPN parameters

41 Building a cosmology with non-linear structures This approach provides: - explicit links between the (post-newtonian) gravitational fields of nonlinear structures on small scales, and the large-scale expansion of space. - a way to relate perturbations to a cosmological RW space, and post- Newtonian perturbations to Minkowski space. dark energy parameters and PPN parameters - plus integrability condition on, and evolution equations for all matter fields. PPNC parameters [Sanghai & TC, CQG 34, (2017)]

42 Building a cosmology with non-linear structures This approach provides: - Information about the scale dependence of slip and effective Newton s constant parameters: from previous slide from consistency conditions

43 [Sanghai & TC, arxiv: ] Building a cosmology with non-linear structures This approach provides: - Information about the scale dependence of slip and effective Newton s constant parameters: Current Observational constraints give: from previous slide from consistency conditions ζ μ 1 1σ confidence regions interpolating tanh functions

44 Building a cosmology with non-linear structures Future work will provide: - Observational constraints on the parameters. (in progress with Bull and Sanghai) - The parameterization of the leading-order vector gravitational potential. (in progress with Thomas and Coates) - Parameterization of non-linear gravitational effects. (in progress)

45 Building a cosmology with non-linear structures Future work will provide: - Observational constraints on the parameters. (in progress with Bull and Sanghai) - The parameterization of the leading-order vector gravitational potential. (in progress with Thomas and Coates) - Parameterization of non-linear gravitational effects. (in progress) may require knowledge of the behaviour of inhomogeneities on horizon-sized scales

46 Gravity on large scales

47 Gravity on large scales If we now consider gravity on the scale of the particle horizon, then our domain of interest looks like this: L C

48 Gravity on large scales If we now consider gravity on the scale of the particle horizon, then our domain of interest looks like this: has no limitations on spatial scale L C

49 Gravity on large scales If we now consider gravity on the scale of the particle horizon, then our domain of interest looks like this: has no limitations on spatial scale L C Requires perturbation theory. and [see e.g. Malik & Wands, Phys. Rept. 475, 1 (2009)]

50 Gravity on large scales If we now consider gravity on the scale of the particle horizon, then our domain of interest looks like this: has no limitations on spatial scale L C Requires perturbation theory. requires small density contrasts and [see e.g. Malik & Wands, Phys. Rept. 475, 1 (2009)]

51 Two-parameter expansions To model structure on all scales we can apply both post-newtonian and cosmological expansions: [Goldberg, TC & Malik, PRD 95, (2017)]

52 Two-parameter expansions To model structure on all scales we can apply both post-newtonian and cosmological expansions: Can be viewed as post-newtonian gravity operating on a perturbed FLRW background. [Goldberg, TC & Malik, PRD 95, (2017)]

53 Two-parameter expansions To model structure on all scales we can apply both post-newtonian and cosmological expansions: Can be viewed as post-newtonian gravity operating on a perturbed FLRW background. Or, equivalently, as cosmological perturbation theory operating on an FLRW background with post- Newtonian perturbations. [Goldberg, TC & Malik, PRD 95, (2017)]

54 Thanks for listening