Editing Primordial fluctuations (section)

==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>