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Primordial fluctuations
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{{More footnotes|date=October 2009}} '''Primordial fluctuations''' are [[density]] variations in the early universe which are considered the seeds of all [[large-scale structure of the cosmos|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 (cosmology)|scale factor]] during inflation caused [[quantum fluctuation]]s of the inflaton field to be stretched to macroscopic scales, and, upon leaving the [[Observable universe#Horizons|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 [[Anisotropy|anisotropies]] in the [[Cosmic microwave background radiation|cosmic microwave background]] and from measurements of the distribution of matter, e.g., galaxy [[redshift survey]]s. Since the fluctuations are believed to arise from inflation, such measurements can also set constraints on parameters within inflationary theory. ==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: :<math>\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}},</math> where <math> \rho </math> is the energy density, <math>\bar{\rho}</math> its average and <math> k </math> the [[wavenumber]] of the fluctuations. The power spectrum <math> \mathcal{P}(k)</math> can then be defined via the ensemble average of the [[Fourier transform|Fourier components]]: :<math> \langle \delta_k \delta_{k'} \rangle = \frac{2 \pi^2}{k^3} \, \delta_D(k-k') \, \mathcal{P}(k),</math> where <math>\delta_D</math> is the [[Dirac delta function]] and angle brackets denote an ensemble average.<ref>https://ned.ipac.caltech.edu/level5/March03/Bertschinger/Bert2_4.html</ref> There are both scalar and tensor modes of fluctuations.{{clarify|why nothing else?|date=March 2017}} ===Scalar modes=== Scalar modes have the power spectrum defined as the mean squared density fluctuation for a specific wavenumber <math>k</math>, i.e., the average fluctuation amplitude at a given scale: :<math>\mathcal{P}_\mathrm{s}(k) = \langle\delta_k\rangle^2.</math> Many inflationary models predict that the scalar component of the fluctuations obeys a [[power law]]{{why|date=March 2017}} in which :<math>\mathcal{P}_\mathrm{s}(k) \propto k^{n_\mathrm{s}}.</math> For scalar fluctuations, <math>n_\mathrm{s}</math> is referred to as the scalar [[spectral index]], with <math>n_\mathrm{s} = 1</math> corresponding to [[scale invariance|scale invariant]] fluctuations (not scale invariant in <math>\delta</math> but in the comoving curvature perturbation <math>\zeta</math> for which the power <math>\mathcal{P}_{\zeta}(k) \propto k^{n_s-1}</math> is indeed invariant with <math>k</math> when <math>n_s=1</math>).<ref>{{cite book|author=Liddle & Lyth |title=Cosmological inflation and large-scale structure |page=75}}</ref> 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 <math>n_s > 1</math>. On the other hand, models such as monomial potentials predict a red spectral index <math>n_s < 1</math>. Planck provides a value of <math>n_s = 0.968 \pm 0.006</math>.<ref name=":0" /> ===Tensor modes=== {{main|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 :<math>r=\frac{2|\delta_h|^2}{|\delta_R|^2},</math> where the 2 arises due to the two polarizations of the tensor modes. 2015 [[cosmic microwave background|CMB]] data from the [[Planck (spacecraft)|Planck satellite]] gives a constraint of <math>r<0.11</math>.<ref name=":0">{{cite journal|page=1 |title=Planck 2015 results. XX. Constraints on inflation|journal=Astronomy & Astrophysics|volume=594|arxiv = 1502.02114|doi = 10.1051/0004-6361/201525898|year = 2016|last1 = Ade|first1 = P. A. R.|last2=Aghanim|first2=N.|author2-link=Nabila Aghanim|last3=Arnaud|first3=M.|last4=Arroja|first4=F.|last5=Ashdown|first5=M.|last6=Aumont|first6=J.|last7=Baccigalupi|first7=C.|last8=Ballardini|first8=M.|last9=Banday|first9=A. J.|last10=Barreiro|first10=R. B.|last11=Bartolo|first11=N.|last12=Battaner|first12=E.|last13=Benabed|first13=K.|last14=Benoît|first14=A.|last15=Benoit-Lévy|first15=A.|last16=Bernard|first16=J.-P.|last17=Bersanelli|first17=M.|last18=Bielewicz|first18=P.|last19=Bock|first19=J. J.|last20=Bonaldi|first20=A.|last21=Bonavera|first21=L.|last22=Bond|first22=J. R.|last23=Borrill|first23=J.|last24=Bouchet|first24=F. R.|last25=Boulanger|first25=F.|last26=Bucher|first26=M.|last27=Burigana|first27=C.|last28=Butler|first28=R. C.|last29=Calabrese|first29=E.|last30=Cardoso|first30=J.-F.|display-authors=29|bibcode=2016A&A...594A..20P|s2cid=119284788}}</ref> ==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== {{Portal|Physics}} *[[Big Bang]] *[[Cosmological perturbation theory]] *[[Cosmic microwave background spectral distortions]] *[[Press–Schechter formalism]] *[[Primordial gravitational wave]] *[[Primordial black hole]] ==References== {{Reflist}} ==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}} * [[Hiranya Peiris|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}} [[Category:Physical cosmology]] [[Category:Inflation (cosmology)]]
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