Estimating temperature fluctuations in the early universe
D. Gangopadhyay^{1}
(1) S. N. Bose National Centre for Basic Sciences, JD-Block, Sector-III, Salt Lake, Kolkata 700098, India
Abstract
A lagrangian for a k-essence field is constructed for a constant scalar potential, and its form is determined when the scale factor is very small as compared to the present epoch but very large as compared to the inflationary epoch. This means that one is already in an expanding and flat universe. The form is similar to that of an oscillator with time-dependent frequency. Expansion is naturally built into the theory with the existence of growing classical solutions of the scale factor. The formalism allows one to estimate the temperature fluctuations of the background radiation at these early stages (as compared to the present epoch) of the Universe. If the temperature is T_{a} at time t_{a} and T_{b} at time t_{b} (t_{b} > t_{a}), then, for small times, the probability evolution for the logarithm of the inverse temperature can be estimated as P(b, a) = | < ln(1/T_{b}), t_{b} | ln(1/T_{a}), t_{a} > |^{2} » (3m_{Pl}^{2} / p^{2} (t_{b} - t_{a})^{3}) (lnT_{a})^{2} (lnT_{b})^{2} (1 - 3g(t_{a} + t_{b})), where 0 < g < 1, m_{Pl} is the Planck mass, and Planck's constant and the speed of light have been put equal to unity. There is a further possibility that a single scalar field may suffice for an inflationary scenario as well as the dark matter and dark energy realms.
References
- D. Gangopadhyay and S. Mukherjee, Logarithm of the scale factor as a generalised coordinate in a Lagrangian for dark matter and dark energy, Phys. Lett. B665, 121 (2008) [arXiv:0710.5366].
- R. J. Scherrer, Purely kinetic k-essence as unified dark matter, Phys. Rev. Lett. 93, 011301 (2004) [astro-ph/040231].
- C. Armendariz-Picon, T. Damour and V. Mukhanov, k-inflation, Phys. Lett. B458, 209 (1999) [hep-th/9904075]; J. Garriga and V. F. Mukhanov, Perturbations in k-inflation, Phys. Lett. B458, 219 (1999) [hep-th/9904176].
- T. Chiba, T. Okabe and M. Yamaguchi, Kinetically driven quintessence, Phys. Rev. D62, 023511 (2000) [astro-ph/9912463]; C. Armendariz-Picon, V. Mukhanov and P. J. Steinhardt, A dynamical solution to the problem of a small cosmological constant and late time cosmic acceleration, Phys. Rev. Lett. 85. 4438 (2000) [astro-ph/0004134]; C. Armendariz-Picon, V. Mukhanov and P. J. Steinhardt, Essentials of k-essence, Phys. Rev. D63, 103510 (2001) [astro-ph/0006373]; T. Chiba, Tracking k-essence, Phys. Rev. D66, 063514 (2002) [astro-ph/0206298].
- L. P. Chimento, Extended tachyon field, Chaplygin gas and solvable k-essence cosmologies, Phys. Rev. D69, 123517 (2004) [astro-ph/0311613].
- M. Gasperini and G. Veneziano, The pre-big bang scenario in string cosmology, Phys. Rep. 373, 1 (2003) [hep-th/0207130]; K.I. Maeda, Towards the Einstein-Hilbert action via conformal transformation, Phys. Rev. D39 3159 (1989).
- V. Sahni, Dark matter and dark energy, Lect. Notes Phys. 653, 141 (2004) [astro-ph/0403324]; 843, 111 (2006) [astro-ph/0602117]; T. Padmanabhan, Dark energy: mystery of the millenium, AIP Conf. Proc. 861, 179 (2006) [astro-ph/0603114]; T. Padmanabhan, Dark energy and gravity, Gen. Rel. Grav. 40, 529 (2007) [arXiv:0705.2533]; E. J. Copeland, M. Sami and S. Tsujikawa, Dynamics of dark energy, Int. J. Mod. Phys. D15, 1753 (2006) [hep-th/0603057].
- P. J. E. Peebles and B. Ratra, The cosmological constant and dark energy, Rev. Mod. Phys. 75, 559 (2003); T. Padmanabhan, Cosmological constant - the weight of the vacuum, Phys. Rep. 380, 235 (2003) [hep-th/0212290].
- M. Malquarti, E. J. Copeland, A. R. Liddle, and M. Trodden, A new view of k-essence, Phys. Rev. D67, 123503 (2003) [astro-ph/0302279]; M. Malquarti, E. J. Copeland, and A. R. Liddle, K-essence and the coincidence problem, Phys. Rev. D68, 023512 (2003) [astro-ph/0304277]; L. Mingzhe and X. Zhang, K-essence leptogenesis, Phys. Lett. B573, 20 (2003) [hep-ph/0209093]; J. M. Aguirregabiria, L. P. Chimento and R. Lazkoz, Phantom k-essence cosmologies, Phys. Rev. D70, 023509 (2004) [astro-ph/0403157].
- L. P. Chimento and R. Lazkoz, Atypical k-essence cosmologies, Phys. Rev. D71, 023505 (2005) [astro-ph/0404494]; L. P. Chimento, M. Forte and R. Lazkoz, Dark matter to dark energy transition in k-essence cosmologies, Mod. Phys. Lett. A20, 2075 (2005) [astro-ph/0407288]; R. Lazkoz, Rigidity of cosmic acceleration in a class of k-essence cosmologies, Int. J. Mod. Phys. D14, 635 (2005) [gr-qc/0410019]; H. Kim, Brans-Dicke scalar field as a unique k-essence, Phys. Lett. B606, 223 (2005) [astro-ph/0408154]; J. M. Aguirregabiria,L. P. Chimento and R. Lazkoz, Quintessence as k-essence, Phys. Lett. B631, 93 (2005) [astro-ph/0411258]; H. Wei and R. G. Cai, K-chamelion and the coincidence problem, Phys. Rev. D71, 043504 (2005) [hep-th/0412045]; C. Armendariz-Picon and E. A. Lim, Haloes of k-essence, JCAP 0508, 7 (2005) [astro-ph/0505207].
- L. R. Abramo and N. Pinto-Neto, On the stability of phantom k-essence theories, Phys. Rev. D73, 063522 (2006) [astro-ph/0511562]; A. D. Rendall, Dynamics of k-essence, Class. Quant. Grav. 23, 1557 (2006) [gr-qc/0511158].
- A. Sen, Rolling tachyon, JHEP 0204, 048 (2002) [hep-th/0203211].
- L. Kofman, A. Linde, and A. Starobinsky, Reheating after inflation, Phys. Rev. Lett. 73, 3195 (1994) [hep-th/9405187]; Towards the theory of reheating after inflation, Phys. Rev. D56, 3258 (1997) [hep-ph/9704452].
- L. D. Landau and E. M. Leifshitz, Mechanics, Vol. 1, Course of Theoretical Physics (Pergamon Press, Oxford, 1976).
- A. Linde, Phys. Lett. 129B, 177 (1983); Particle Physics and Inflationery Cosmology (Harwood, Chur, 1990).
- Shinji Tsujikawa, Introductory Review of Cosmic Inflation [hep-ph/0304257].
- A. R. Liddle and D. H. Lyth, in Cosmological Inflation and Large Scale Structure (Cambridge University Press, 2000).
- D. C. Khandekar and S. V. Lawande, Phys. Rep. 137, 115 (1986).
- H. Ezawa, J. R. Klauder, and L. A. Shepp, J. Math. Phys. 16, 783 (1975).
- B. Simon, J. Functional Analysis and Applications 14, 295 (1973).
- T. E. Clark, R. Menikoff and D. H. Sharp, Phys. Rev. D22, 3012 (1980).
- R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw Hill, New York, 1965).
- V. P. Ermakov, Univ. Izv. Kiev 20, 1 (1880); E. Pinney, Proc. Am. Math. Soc. 1, 681 (1950).
For more information about this paper please visit Springer's Home Page of this paper.
Back to The Contents Page