# A general proof of the conservation of the curvature perturbation

@article{Lyth2005AGP, title={A general proof of the conservation of the curvature perturbation}, author={David H. Lyth and Karim A. Malik and Misao Sasaki}, journal={Journal of Cosmology and Astroparticle Physics}, year={2005}, volume={2005}, pages={004 - 004} }

Without invoking a perturbative expansion, we define the cosmological curvature perturbation, and consider its behaviour assuming that the universe is smooth over a sufficiently large comoving scale. The equations are simple, resembling closely the first-order equations, and they lead to results which generalize those already proven in linear perturbation theory and (in part) in second-order perturbation theory. In particular, the curvature perturbation is conserved provided that the pressure…

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## References

SHOWING 1-10 OF 43 REFERENCES

### A New approach to the evolution of cosmological perturbations on large scales

- Physics
- 2000

We discuss the evolution of linear perturbations about a Friedmann-Robertson-Walker background metric, using only the local conservation of energy-momentum. We show that on sufficiently large scales…

### Conserved cosmological perturbations

- Physics, Mathematics
- 2003

A conserved cosmological perturbation is associated with each quantity whose local evolution is determined entirely by the local expansion of the Universe. It may be defined as the appropriately…

### Gauge Invariant Cosmological Perturbations

- Physics
- 1980

The physical interpretation of perturbations of homogeneous, isotropic cosmological models in the early Universe, when the perturbation is larger than the particle horizon, is clarified by defining a…

### Separate universe approach and the evolution of nonlinear superhorizon cosmological perturbations

- Physics
- 2003

In this paper we review the separate universe approach for cosmological perturbations and point out that it is essentially the lowest order approximation to a gradient expansion. Using this approach,…

### Evolution of second-order cosmological perturbations.

- Physics, Mathematics
- 2003

We present a method for constructing gauge-invariant cosmological perturbations which are gauge-invariant up to second order. As an example, we give the gauge-invariant definition of the second-order…

### Perturbations of spacetime: gauge transformations and gauge invariance at second order and beyond

- Mathematics, Physics
- 1997

We consider in detail the problem of gauge dependence that exists in relativistic perturbation theory, going beyond the linear approximation and treating second- and higher-order perturbations. We…

### BACKREACTION PROBLEM FOR COSMOLOGICAL PERTURBATIONS

- Physics
- 1997

We derive the effective energy-momentum tensor for cosmological perturbations and prove its gauge-invariance. The result is applied to study the influence of perturbations on the behaviour of the…

### Quasi-isotropic solution of the Einstein equations near a cosmological singularity for a two-fluid cosmological model

- Physics
- 2003

The quasi-isotropic inhomogeneous solution of the Einstein equations near a cosmological singularity in the form of a series expansion in the synchronous system of reference, first found by Lifshitz…

### Comment about quasi-isotropic solution of Einstein equations near the cosmological singularity

- Physics, Mathematics
- 2002

For the case of arbitrary hydrodynamical matter, we generalize the quasi-isotropic solution of Einstein equations near the cosmological singularity, found by Lifshitz and Khalatnikov in 1960 for the…

### Second-order perturbations of the Friedmann world model

- Physics
- 2004

We consider the instability of the Friedmann world model to second order in perturbations. We present the perturbed set of equations up to second order in the Friedmann background world model with a…