Primordial fluctuations are density variations in the early universe which are considered the seeds of all structure in the universe. Currently, the most widely accepted explanation for their origin is in the context of cosmic inflation. According to the inflationary paradigm, the exponential growth of the scale factor during inflation caused quantum fluctuations of the inflation field to be stretched to macroscopic scales, and, upon leaving the horizon, to “freeze in”. At the later stages of radiation- and matter-domination, these fluctuations re-entered the horizon, and thus set the initial conditions for structure formation.
The statistical properties of the primordial fluctuations can be inferred from observations of anisotropies in the cosmic microwave background and from measurements of the distribution of matter, e.g., galaxy redshift surveys. Since the fluctuations are believed to arise from inflation, such measurements can also set constraints on parameters within inflationary theory.
Contents
1 Formalism
1.1 Scalar modes
1.2 Tensor modes
2 Adiabatic/isocurvature fluctuations
3 See also
4 References
5 External links
Formalism
Primordial fluctuations are typically quantified by a power spectrum which gives the power of the variations as a function of spatial scale. Within this formalism, one usually considers the fractional energy density of the fluctuations, given by:
{\displaystyle \delta ({\vec {x}})\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\rho ({\vec {x}})}{\bar {\rho }}}-1=\int {\text{d}}k\;\delta _{k}\,e^{i{\vec {k}}\cdot {\vec {x}}},}\delta(\vec{x}) \ \stackrel{\mathrm{def}}{=}\ \frac{\rho(\vec{x})}{\bar{\rho}} – 1 =
\int \text{d}k \; \delta_k \, e^{i\vec{k} \cdot \vec{x}},
where {\displaystyle \rho }\rho is the energy density, {\displaystyle {\bar {\rho }}}{\bar {\rho }} its average and {\displaystyle k}k the wavenumber of the fluctuations. The power spectrum {\displaystyle {\mathcal {P}}(k)} \mathcal{P}(k) can then be defined via the ensemble average of the Fourier components:
{\displaystyle \langle \delta _{k}\delta _{k’}\rangle ={\frac {2\pi ^{2}}{k^{3}}}\,\delta (k-k’)\,{\mathcal {P}}(k).} \langle \delta_k \delta_{k’} \rangle = \frac{2 \pi^2}{k^3} \, \delta(k-k’) \, \mathcal{P}(k).
There are both scalar and tensor modes of fluctuations.[clarification needed]
Scalar modes
Scalar modes have the power spectrum
{\displaystyle {\mathcal {P}}_{\mathrm {s} }(k)=|\delta _{R}|^{2}.}{\displaystyle {\mathcal {P}}_{\mathrm {s} }(k)=|\delta _{R}|^{2}.}[clarification needed]
Many inflationary models predict that the scalar component of the fluctuations obeys a power law[why?] in which
{\displaystyle {\mathcal {P}}_{\mathrm {s} }(k)\propto k^{n_{\mathrm {s} }-1}.}\mathcal{P}_\mathrm{s}(k) \propto k^{n_\mathrm{s} – 1}.
For scalar fluctuations, {\displaystyle n_{\mathrm {s} }}n_\mathrm{s} is referred to as the scalar spectral index, with {\displaystyle n_{\mathrm {s} }=1}n_\mathrm{s} = 1 corresponding to scale invariant fluctuations.[1]
The scalar spectral index describes how the density fluctuations vary with scale. As the size of these fluctuations depends upon the inflaton’s motion when these quantum fluctuations are becoming super-horizon sized, different inflationary potentials predict different spectral indices. These depend upon the slow roll parameters, in particular the gradient and curvature of the potential. In models where the curvature is large and positive {\displaystyle n_{s}>1}{\displaystyle n_{s}>1}. On the other hand, models such as monomial potentials predict a red spectral index {\displaystyle n_{s}<1}{\displaystyle n_{s}<1}. Planck provides a value of {\displaystyle n_{s}}n_{s} of 0.96.
Tensor modes
Main article: Gravitational wave
The presence of primordial tensor fluctuations is predicted by many inflationary models. As with scalar fluctuations, tensor fluctuations are expected to follow a power law and are parameterized by the tensor index (the tensor version of the scalar index). The ratio of the tensor to scalar power spectra is given by
{\displaystyle r={\frac {2|\delta _{h}|^{2}}{|\delta _{R}|^{2}}},}{\displaystyle r={\frac {2|\delta _{h}|^{2}}{|\delta _{R}|^{2}}},}
where the 2 arises due to the two polarizations of the tensor modes. 2015 CMB data from the Planck satellite gives a constraint of {\displaystyle r<0.11}{\displaystyle r<0.11}.[2]
Adiabatic/isocurvature fluctuations
Adiabatic fluctuations are density variations in all forms of matter and energy which have equal fractional over/under densities in the number density. So for example, an adiabatic photon overdensity of a factor of two in the number density would also correspond to an electron overdensity of two. For isocurvature fluctuations, the number density variations for one component do not necessarily correspond to number density variations in other components. While it is usually assumed that the initial fluctuations are adiabatic, the possibility of isocurvature fluctuations can be considered given current cosmological data. Current cosmic microwave background data favor adiabatic fluctuations and constrain uncorrelated isocurvature cold dark matter modes to be small.
See also
icon Physics portal
Big Bang
Cosmological perturbation theory
Cosmic microwave background spectral distortions
Press–Schechter formalism
Primordial gravitational wave
Primordial black hole
References
Liddle & Lyth. Cosmological inflation and large-scale structure. p. 75.
Ade, P. A. R.; Aghanim, N.; Arnaud, M.; Arroja, F.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Ballardini, M.; Banday, A. J.; Barreiro, R. B.; Bartolo, N.; Battaner, E.; Benabed, K.; Benoît, A.; Benoit-Lévy, A.; Bernard, J.-P.; Bersanelli, M.; Bielewicz, P.; Bock, J. J.; Bonaldi, A.; Bonavera, L.; Bond, J. R.; Borrill, J.; Bouchet, F. R.; Boulanger, F.; Bucher, M.; Burigana, C.; Butler, R. C.; Calabrese, E.; et al. (2016). "Planck 2015 results. XX. Constraints on inflation". Astronomy & Astrophysics. 594: 1. arXiv:1502.02114. Bibcode:2016A&A…594A..20P. doi:10.1051/0004-6361/201525898. S2CID 119284788.
External links
Crotty, Patrick, "Bounds on isocurvature perturbations from CMB and LSS data". Physical Review Letters. arXiv:astro-ph/0306286
Linde, Andrei, "Quantum Cosmology and the Structure of Inflationary Universe". Invited talk. arXiv:gr-qc/9508019
Peiris, Hiranya, "First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Implications for Inflation". Astrophysical Journal. arXiv:astro-ph/0302225
Tegmark, Max, "Cosmological parameters from SDSS and WMAP". Physical Review D. arXiv:astro-ph/0310723
Categories: Physical cosmologyInflation (cosmology)
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i dont konw, jerell, how?
Primordial fluctuations are density variations in the early universe which are considered the seeds of all structure in the universe. Currently, the most widely accepted explanation for their origin is in the context of cosmic inflation. According to the inflationary paradigm, the exponential growth of the scale factor during inflation caused quantum fluctuations of the inflation field to be stretched to macroscopic scales, and, upon leaving the horizon, to “freeze in”. At the later stages of radiation- and matter-domination, these fluctuations re-entered the horizon, and thus set the initial conditions for structure formation.
The statistical properties of the primordial fluctuations can be inferred from observations of anisotropies in the cosmic microwave background and from measurements of the distribution of matter, e.g., galaxy redshift surveys. Since the fluctuations are believed to arise from inflation, such measurements can also set constraints on parameters within inflationary theory.
Contents
1 Formalism
1.1 Scalar modes
1.2 Tensor modes
2 Adiabatic/isocurvature fluctuations
3 See also
4 References
5 External links
Formalism
Primordial fluctuations are typically quantified by a power spectrum which gives the power of the variations as a function of spatial scale. Within this formalism, one usually considers the fractional energy density of the fluctuations, given by:
{\displaystyle \delta ({\vec {x}})\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\rho ({\vec {x}})}{\bar {\rho }}}-1=\int {\text{d}}k\;\delta _{k}\,e^{i{\vec {k}}\cdot {\vec {x}}},}\delta(\vec{x}) \ \stackrel{\mathrm{def}}{=}\ \frac{\rho(\vec{x})}{\bar{\rho}} – 1 =
\int \text{d}k \; \delta_k \, e^{i\vec{k} \cdot \vec{x}},
where {\displaystyle \rho }\rho is the energy density, {\displaystyle {\bar {\rho }}}{\bar {\rho }} its average and {\displaystyle k}k the wavenumber of the fluctuations. The power spectrum {\displaystyle {\mathcal {P}}(k)} \mathcal{P}(k) can then be defined via the ensemble average of the Fourier components:
{\displaystyle \langle \delta _{k}\delta _{k’}\rangle ={\frac {2\pi ^{2}}{k^{3}}}\,\delta (k-k’)\,{\mathcal {P}}(k).} \langle \delta_k \delta_{k’} \rangle = \frac{2 \pi^2}{k^3} \, \delta(k-k’) \, \mathcal{P}(k).
There are both scalar and tensor modes of fluctuations.[clarification needed]
Scalar modes
Scalar modes have the power spectrum
{\displaystyle {\mathcal {P}}_{\mathrm {s} }(k)=|\delta _{R}|^{2}.}{\displaystyle {\mathcal {P}}_{\mathrm {s} }(k)=|\delta _{R}|^{2}.}[clarification needed]
Many inflationary models predict that the scalar component of the fluctuations obeys a power law[why?] in which
{\displaystyle {\mathcal {P}}_{\mathrm {s} }(k)\propto k^{n_{\mathrm {s} }-1}.}\mathcal{P}_\mathrm{s}(k) \propto k^{n_\mathrm{s} – 1}.
For scalar fluctuations, {\displaystyle n_{\mathrm {s} }}n_\mathrm{s} is referred to as the scalar spectral index, with {\displaystyle n_{\mathrm {s} }=1}n_\mathrm{s} = 1 corresponding to scale invariant fluctuations.[1]
The scalar spectral index describes how the density fluctuations vary with scale. As the size of these fluctuations depends upon the inflaton’s motion when these quantum fluctuations are becoming super-horizon sized, different inflationary potentials predict different spectral indices. These depend upon the slow roll parameters, in particular the gradient and curvature of the potential. In models where the curvature is large and positive {\displaystyle n_{s}>1}{\displaystyle n_{s}>1}. On the other hand, models such as monomial potentials predict a red spectral index {\displaystyle n_{s}<1}{\displaystyle n_{s}<1}. Planck provides a value of {\displaystyle n_{s}}n_{s} of 0.96.
Tensor modes
Main article: Gravitational wave
The presence of primordial tensor fluctuations is predicted by many inflationary models. As with scalar fluctuations, tensor fluctuations are expected to follow a power law and are parameterized by the tensor index (the tensor version of the scalar index). The ratio of the tensor to scalar power spectra is given by
{\displaystyle r={\frac {2|\delta _{h}|^{2}}{|\delta _{R}|^{2}}},}{\displaystyle r={\frac {2|\delta _{h}|^{2}}{|\delta _{R}|^{2}}},}
where the 2 arises due to the two polarizations of the tensor modes. 2015 CMB data from the Planck satellite gives a constraint of {\displaystyle r<0.11}{\displaystyle r<0.11}.[2]
Adiabatic/isocurvature fluctuations
Adiabatic fluctuations are density variations in all forms of matter and energy which have equal fractional over/under densities in the number density. So for example, an adiabatic photon overdensity of a factor of two in the number density would also correspond to an electron overdensity of two. For isocurvature fluctuations, the number density variations for one component do not necessarily correspond to number density variations in other components. While it is usually assumed that the initial fluctuations are adiabatic, the possibility of isocurvature fluctuations can be considered given current cosmological data. Current cosmic microwave background data favor adiabatic fluctuations and constrain uncorrelated isocurvature cold dark matter modes to be small.
See also
icon Physics portal
Big Bang
Cosmological perturbation theory
Cosmic microwave background spectral distortions
Press–Schechter formalism
Primordial gravitational wave
Primordial black hole
References
Liddle & Lyth. Cosmological inflation and large-scale structure. p. 75.
Ade, P. A. R.; Aghanim, N.; Arnaud, M.; Arroja, F.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Ballardini, M.; Banday, A. J.; Barreiro, R. B.; Bartolo, N.; Battaner, E.; Benabed, K.; Benoît, A.; Benoit-Lévy, A.; Bernard, J.-P.; Bersanelli, M.; Bielewicz, P.; Bock, J. J.; Bonaldi, A.; Bonavera, L.; Bond, J. R.; Borrill, J.; Bouchet, F. R.; Boulanger, F.; Bucher, M.; Burigana, C.; Butler, R. C.; Calabrese, E.; et al. (2016). "Planck 2015 results. XX. Constraints on inflation". Astronomy & Astrophysics. 594: 1. arXiv:1502.02114. Bibcode:2016A&A…594A..20P. doi:10.1051/0004-6361/201525898. S2CID 119284788.
External links
Crotty, Patrick, "Bounds on isocurvature perturbations from CMB and LSS data". Physical Review Letters. arXiv:astro-ph/0306286
Linde, Andrei, "Quantum Cosmology and the Structure of Inflationary Universe". Invited talk. arXiv:gr-qc/9508019
Peiris, Hiranya, "First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Implications for Inflation". Astrophysical Journal. arXiv:astro-ph/0302225
Tegmark, Max, "Cosmological parameters from SDSS and WMAP". Physical Review D. arXiv:astro-ph/0310723
Categories: Physical cosmologyInflation (cosmology)
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Languages
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Català
Deutsch
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Edit links
This page was last edited on 5 April 2022, at 21:59 (UTC).
Text is available under the Creative Commons Attribution-ShareAlike License 3.0; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.