1. A. Woszczyna, W. Czaja, K. Głód, Z. A. Golda, R. A. Kycia, A. Odrzywołek, P. Plaszczyk, L. M. Sokołowski, S. J. Szybka
ccgrg: The symbolic tensor analysis package with tools for general relativity
Wolfram Library Archive, vol. 8848 (2014).
[abstract] [journal]

Abstract:
Riemann and Weyl curvature, covariant derivative, Lie derivative, the first and the second fundamental form on hyper-surfaces, as well as basic notions of relativistic hydrodynamics (expansion, vorticity, shear) are predefined functions of the package. New tensors are easy to define. Instructions, basic examples, and some more advanced examples are attached to the package. Characteristic feature of the ccgrg package is the specific coupling between the functional programming and the Parker-Christensen index convention. This causes that no particular tools to rising/lowering tensor indices neither to the tensor contractions are needed. Tensor formulas are written in the form close to that of classical textbooks in GRG, with the only difference that the summation symbol appears explicitly. Tensors are functions, not matrixes, and their components are evaluated lazily. This means that only these components which are indispensable to realize the final task are computed. The memoization technique prevents repetitive evaluation of the same quantities. This saves both, time and memory.

2. Wojciech Czaja, Zdzislaw A. Golda, Andrzej Woszczyna
The acoustic wave equation in the expanding universe. Sachs-Wolfe theorem
The Mathematica Journal, vol. 13 (2011).
[abstract] [journal] [cdf]

Abstract:
This article considers the acoustic field propagating in the radiation-dominated universe of arbitrary space curvature. The field equations are reduced to the d’Alembert equation in an auxiliary static Robertson-Walker spacetime and dispersion relations are discussed. *supported by the grant from The John Templeton Foundation

3. Marek Demianski, Zdzislaw A. Golda, Andrzej Woszczyna
Evolution of density perturbations in a realistic universe
Gen. Rel. Grav., vol. 37, pp. 2063-2082 (2005).
[abstract] [preprint] [journal]

Abstract:
Prompted by the recent more precise determination of the basic cosmological parameters and growing evidence that the matter-energy content of the universe is now dominated by dark energy and dark matter we present the general solution of the equation that describes the evolution of density perturbations in the linear approximation. It turns out that as in the standard CDM model the density perturbations grow very slowly during the radiation dominated epoch and their amplitude increases by a factor of about 4000 in the matter and later dark energy dominated epoch of expansion of the universe.

4. Zdzisław A. Golda, Andrzej Woszczyna, Karolina Zawada
Canonical gauge-invariant variables
Acta Phys. Pol., B , vol. B36, p. 2133 (2005).
[abstract] [preprint] [journal]

Abstract:
Under an appropriate change of the perturbation variable Lifshitz-Khalatnikov propagation equations for the scalar perturbation reduce to d'Alembert equation. The change of variables is based on the Darboux transform

5. G. Siemieniec-Ozieblo, A. Woszczyna
Acoustic instabilities at the transition from the radiation-dominated to the matter-dominated universe
Astron. Astrophys. , vol. 419, pp. 801-810 (2004).
[abstract] [preprint] [journal]

Abstract:
The transition from acoustic noise in the radiation-dominated universe to the density structures in the matter dominated epoch is considered. The initial state is a stochastic field of sound waves moving in different directions. The construction of the initial state is compatible with the hyperbolic type of propagation equation for density perturbations, and parallel to the theory of stochastic background of gravitational waves. Instantaneous transition between the cosmological epochs is assumed, and Darmois-Israel joining conditions are applied to match solutions for sound waves with growing or decaying modes at the decoupling. As a result a substantial amplification of the low scale structures is obtained.

6. Z. Golda, A.Woszczyna
Dispersion of density waves in the early universe with positive cosmological constant
Class. Quantum Grav., vol. 20, p. 277 (2003).
[abstract] [preprint] [journal]

Abstract:
Density perturbations in the flat (K=0) Robertson-Walker universe with radiation ($p=\epsilon/3$) and positive cosmological constant ($\Lambda>0$) are investigated. The phenomenon of anomalous dispersion of acoustic waves on $\Lambda$ is discussed.