Primordial fluctuations are

density Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...

variations in the early universe which are considered the seeds of all

structure A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...

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 inflaton field to be stretched to macroscopic scales, and, upon leaving the

horizon The horizon is the apparent curve that separates the surface of a celestial body from its sky when viewed from the perspective of an observer on or near the surface of the relevant body. This curve divides all viewing directions based on whethe ...

, 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 The cosmic microwave background (CMB, CMBR), or relic radiation, is microwave radiation that fills all space in the observable universe. With a standard optical telescope, the background space between stars and galaxies is almost completely dar ...

and from measurements of the distribution of matter, e.g., galaxy

redshift survey In astronomy, a redshift survey is a astronomical surveys, survey of a section of the sky to measure the redshift of astronomical objects: usually galaxies, but sometimes other objects such as galaxy clusters or quasars. Using Hubble's law, the ...

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 In signal processing, the power spectrum S_(f) of a continuous time signal x(t) describes the distribution of Power (physics), power into frequency components f composing that signal. According to Fourier analysis, any physical signal can be ...

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: :

\delta(\vec) \ \stackrel\   \frac - 1 =
 \int \textk \; \delta_k \, e^,

where

\rho

is the energy density,

\bar

its average and

k

the

wavenumber In the physical sciences, the wavenumber (or wave number), also known as repetency, is the spatial frequency of a wave. Ordinary wavenumber is defined as the number of wave cycles divided by length; it is a physical quantity with dimension of ...

of the fluctuations. The power spectrum

\mathcal(k)

can then be defined via the ensemble average of the Fourier components: :

\langle \delta_k \delta_ \rangle = \frac \, \delta_D(k-k') \, \mathcal(k),

where

\delta_D

is the

Dirac delta function In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...

and angle brackets denote an ensemble average. There are both scalar and tensor modes of fluctuations.

Scalar modes

Scalar modes have the power spectrum defined as the mean squared density fluctuation for a specific wavenumber

k

, i.e., the average fluctuation amplitude at a given scale: :

\mathcal_\mathrm(k) = \langle\delta_k\rangle^2.

Many inflationary models predict that the scalar component of the fluctuations obeys a

power law In statistics, a power law is a Function (mathematics), functional relationship between two quantities, where a Relative change and difference, relative change in one quantity results in a relative change in the other quantity proportional to the ...

in which :

\mathcal_\mathrm(k) \propto k^.

For scalar fluctuations,

n_\mathrm

is referred to as the scalar spectral index, with

n_\mathrm = 1

corresponding to scale invariant fluctuations (not scale invariant in

\delta

but in the comoving curvature perturbation

\zeta

for which the power

\mathcal_(k) \propto k^

is indeed invariant with

k

when

n_s=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

n_s > 1

. On the other hand, models such as monomial potentials predict a red spectral index

n_s < 1

. Planck provides a value of

n_s = 0.968 \pm 0.006

Tensor modes

The presence of primordial

tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...

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 :

r=\frac,

where the 2 arises due to the two polarizations of the tensor modes. 2015 CMB data from the Planck satellite gives a constraint of

r<0.11

Adiabatic/isocurvature fluctuations

Adiabatic fluctuations are density variations in all forms of

matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic pa ...

and

energy Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...

which have equal fractional over/under densities in the number density. So for example, an adiabatic

photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that can ...

overdensity of a factor of two in the number density would also correspond to an

electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...

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

data favor adiabatic fluctuations and constrain uncorrelated isocurvature cold dark matter modes to be small.

References

External links