# Cosmological Sakharov Oscillations and Quantum Mechanics of the Early Universe

###### Abstract

This is a brief summary of a talk delivered at the Special Session of the Physical Sciences Division of the Russian Academy of Sciences, Moscow, 25 May 2011. The meeting was devoted to the 90-th anniversary of the birth of A. D. Sakharov. The focus of this contribution is on the standing-wave pattern of quantum-mechanically generated metric (gravitational field) perturbations as the origin of subsequent Sakharov oscillations in the matter power spectrum. Other related phenomena, particularly in the area of gravitational waves, and their observational significance are also discussed.

###### pacs:

98.70.Vc, 98.80.Cq, 04.30.-w## I Sakharov’s first cosmological paper

The ideas and results of Andrei Sakharov’s remarkable paper Sakh65 have influenced the course of cosmological research and are still in the centre of theoretical and observational studies. The title of his paper was “The initial stage of an expanding universe and the appearance of a nonuniform distribution of matter”. The paper was submitted to ZhETF on 2 March 1965, that is, in the days when not only the existence of the cosmic microwave background radiation (CMB) was not yet established, but even the nonstationarity of the Universe was still debated. Right the second sentence of the Abstract says: “It is assumed that the initial inhomogeneities arise as a result of quantum fluctuations of cold baryon-lepton matter at densities of the order of baryons/cm. It is suggested that at such densities gravitational effects are of decisive importance in the equation of state…”.

In what follows, we discuss recent attempts to explain the appearance of cosmological perturbations (density inhomogeneities, gravitational waves, and possibly rotational perturbations) as a result of quantum processes. In our approach, the perturbations arise as a consequence of superadiabatic (parametric) amplification of quantum-mechanical fluctuations of the appropriate degrees of freedom of the gravitational field itself. So, for us, gravity is of decisive importance not so much because of its contribution to the equation of state of the primeval matter, but because the gravitational field (metric) perturbations are the primary object of quantization. Nevertheless, it must be stressed that the mind-boggling idea suggesting that something microscopic and quantum-mechanical can be responsible for the emergence of fields and observed structures at astronomical scales was first formulated and partially explored in Sakharov’s paper.

A considerable part of the paper Sakh65 is devoted to the evolution
of small density perturbations, rather than to their origin.
The spatial Fourier component of the relative density perturbation
is denoted , where is
a wavenumber. The function satisfies a second-order
differential equation, numbered in the paper as Eq.(15), which follows from
the perturbed Einstein equations. The calculation leading to the phenomenon
which was later named the Sakharov oscillations is introduced by the following
words:

Yu. M. Shustov and V. A. Tarasov have at our request solved Eq.(15), with
the aid of an electronic computer, for different values of . The
calculations were made for the simplest equation of state, satisfying
with and with
(A is a constant )

In the paper Sakh65 , the quantity , , is the particle number density, is the energy density in the rest frame of the material, and is the pressure. Obviously, the interpolating formula (16) describes the transition from the relativistic equation of state , applicable at early times of evolution and relatively large , to the nonrelativistic equation of state valid at small and late times. During the transition, the speed of sound decreases from to .

It is important to realize that the physical nature of the discussed transition from to can be quite general. Being guided by physical assumptions of his time, Sakharov speaks about cold baryon-lepton matter, degenerate Fermi gas of relativistic noninteracting particles, and so on. But it is important to remember that the perturbed Einstein equations, such as Eq.(15), do not require knowledge of microscopic causes of elasticity and associated speed of sound. Gravitational equations operate with the energy-momentum tensor of the material and its bulk mechanical properties, such as the energy density, pressure, and the link between them, the equation of state. These are postulated by Eq.(16), and one can now think of the results of the performed calculation as a qualitative model of what can happen in other transitions. For example, in a transition from the fluid dominated by photon gas with the equation of state to the fluid dominated by cold dark matter (CDM) with the equation of state .

For simple models of matter, such as and
, Eq.(15) can be solved in elementary functions.
Sakharov writes:

When , the solution of
this equation is expressed in terms of Bessel functions; for example, when
we have increasing and decreasing solutions of the form
()

Indeed, these are the well known solutions for in the medium. The general solution to Eq.(15) is a linear combination of these two branches with arbitrary (in general, complex) coefficients. The first solution can be called increasing and the second decreasing, because at very small they behave as and , respectively. At later times, not long before the transition to the regime, the functions represent ordinary acoustic waves with the oscillatory time dependence and . One does not learn anything new from matching the increasing/decreasing part of a solution to the oscillatory part of the same solution; the general solution is already given by the formula above. At the stage the solutions for do not oscillate as functions of time; they are power-law functions of .

The crucial observation in the Sakharov paper is contained in the following
quotation:

The function can be obtained in the case of Eq.(16) analytically
(Shustov). Shustov and Tarasov find, by integrating (15) the limiting value
as of the auxilliary variable

putting as . It is obvious that
.

In accordance with the results of the sections
that follow, we put . is a function of the
parameter . This function is oscillating and sign-alternating,
but attenuates rapidly with increasing .

The last sentence of this quotation is a surprising statement of incredible importance. It says that well after the transition to the regime () the density fluctuation becomes an oscillating and sign-alternating function of the wavenumber . The square of this function is what can be called a power spectrum. Sakharov uses at the very early times and takes as from his quantum-mechanical considerations. So, it is stated that the initial smooth power spectrum transforms into an oscillatory final power spectrum which has a series of zeros and maxima at some specific wavenumbers . If one imagines that in the era before the transition to the regime the field of sound waves was represented by a set of harmonic oscillators with different frequencies, then the claim is that well after the transition some oscillators will find themselves “lucky”, in the sense that they appear in the maxima of the resulting power spectrum, while others - “unlucky”, because they are at the zeros of the resulting power spectrum.

Certainly, such a striking conclusion cannot be unconditionally true. After all, a computer can be asked to make a similar calculation, but backwards in time. In this calculation, one can postulate a smooth power spectrum at the late stage and evolve the spectrum back in time to derive the functions at the early stage. The derived functions will not coincide with what was taken as initial conditions in the original calculation Sakh65 , but such new initial conditions are possible in principle. By construction, these new initial conditions would not lead to the final power spectrum oscillations. On the other hand, if the oscillations do arise from physically justified initial conditions, then this is an extremely important phenomenon. It dictates the appearance of a periodic structure in Fourier space (a “standard ruler” with characteristic spatial scales) which can be recognized in observations and can be used as a tool for other measurements.

The point of this remark is to stress that, as will be argued below in more detail, the initial conditions leading to Sakharov oscillations are inevitable, if the primordial cosmological perturbations were indeed generated quantum-mechanically.

The oscillatory transfer function participates in further calculations Sakh65 , but it takes quite a modest role there. Sakharov himself did not elaborate on the discovered phenomenon in later publications. However, it seems to me that he was perfectly well aware of the importance of his observation, and he attentively followed subsequent developments. Some evidence for this will be given in sec.IV.

It was Ya.B.Zeldovich who assigned significant value to the discovered oscillations and named them the Sakharov oscillations. In conversations, at seminars, in papers with R.A.Sunyaev, A.G.Doroshkevich, and in a book with I.D.Novikov, Zeldovich discussed the physics of the phenomenon and its possible observational applications. Zeldovich and coauthors deserve credit for seeing the relevance of Sakharov’s work for their own studies and for mentioning his paper. For example, one of the first papers on the subject in the context of a “hot” model of the Universe SZ70 remarks: “at a later stage of expansion the amplitude of density perturbations turns out to be a periodic function of a wavelength (mass). Such a picture was previously obtained by Sakharov (1965) for a cold model of the Universe”. And more SZ70 : “The picture presented above is only a rough approximation since the phase relations between density and velocity perturbations in standing waves in an ionized plasma were not considered. As mentioned in the introduction, Sakharov (1965) showed that the amplitude of perturbations of matter at a later stage when pressure does not play a role (in our case after recombination) turns out to be a periodic function of wavelength”. Zeldovich and Novikov ZN83 discuss the phenomenon at some length and note that “The distribution of astronomical objects with respect to mass will thus reflect the Sakharov oscillations in a very smoothed-out form only. It is possible that they may not be noticed in a study of the mass spectrum”. Fortunately, as we shall see below, there was a significant observational progress in revealing Sakharov oscillations.

The parallel to SZ70 and more detailed paper by P.J.Peebles and J.Yu PY70 explicitly presents in Fig.5 a modulated spectrum, with maxima and zeros, and mentions the relevance of the “first big peak in Fig.5” to the future experimental searches of irregularities in the microwave background radiation. The spectral modulation was derived as a result of numerical calculations. The later private correspondence on the physical interpretation of oscillations inevitably ended up with “lucky” and “unlucky” oscillators Peebles90 : “The Sakharov oscillations you mention also were considered by Jer Yu and me (a few years after Sakarov)…. Here there truly are modes that are unlucky, in the sense that they carry negligible energy”.

To better understand the Sakharov oscillations, as well as other closely related phenomena, we have to make some formalization of the problem. We will do this in the next section. Before that, it is interesting to note, as a side remark, that in course of his quantum-mechanical considerations Sakharov discusses the “initial stage of the expansion of the universe”, and in particular with the scale factor as . He found this evolution in two cases, and , out of the four considered. This type of the scale factor is now advertized as inflation. However, Sakharov himself was sceptical about cases and . He finds arguments against them and concludes: “For these reasons we turn to curves and ”. (Criticism of contemporary inflationary claims can be found in GrWhel , GrComm .)

## Ii Wave-fields of different nature in time-dependent environments

The main physical reason behind Sakharov oscillations, and indeed behind many other similar phenomena, is the time-dependence of the parameters characterizing the environment in which a wave-field is given. This can be a changing speed of sound, or a changing background gravitational field, or all such factors together. In cosmology, the central object is the gravitational field (metric) perturbations. Other quantities, such as fluctuations in density and velocity of matter (if they are present; we recall that they are absent in the case of gravitational waves), are calculable from the metric perturbations via the perturbed Einstein equations. It is only in special conditions and for relatively short-scale variations that the gravitational field perturbations can be neglected.

The gravitational field perturbation is defined by

(1) |

For each of the three types of cosmological perturbations (density perturbations, gravitational waves, and rotational perturbations) the field can be expanded over spatial Fourier modes with wavevectors :

(2) |

The power spectrum (variance) of a given field is a quadratic combination of the field averaged over space, or over known classical probability density function, or over known quantum-mechanical state. In all cases, one arrives at the expression of the following structure

(3) |

The quantity

(4) |

is called the metric power spectrum. At each instance of time, the metric power spectrum is determined by the absolute value of the (in general, – complex) gravitational mode functions . (We often suppress the index , which marks two polarisiation states present in metric perturbations of each type of cosmological perturbations.) For calculation of power spectra of other quantities participating in the problem, one has to expand these quantities as in Eq.(2) and then use their mode functions in expressions for their power spectra, similar to Eq.(4).

The gravitational mode functions , as well as mode functions of other quantities participating in our problem, satisfy one or another version of the second-order differential “master equation” BG

(5) |

where the “speed of sound” and the “potential” are, in general, functions of time. In particular, the Sakharov mode functions for density perturbations obey a specific equation of this kind (written in the -time). And the above-quoted Sakharov solution, for , expressed in terms of Bessel functions with the argument is a particular case in which , whereas is a simple function of the scale factor . Gravitational wave equations are also equations of this form with .

Two linearly-independent high-frequency solutions (i.e. solutions of “master equation” (5) without and with ) are usually taken as . If these mode functions represent sound waves not long before the transition to the regime, then using them for calculation of the power spectrum one would find and, hence, the absence of oscillations in the power spectrum of density perturbations. Therefore, we do not expect any segregation into “lucky” and “unlucky” oscillators in the post-transition era. The general decomposition (2) should be looked at more closely.

The general high-frequency solution to Eq.(5) (for simplicity, we set temporarily ) is , where complex coefficients are in general arbitrary functions of . The -mode of the field

is a sum of two waves traveling in opposite directions with arbitrary amplitudes and arbitrary phases. One particular traveling wave is chosen by setting or . In contrast, the choice makes the field a standing wave, that is, a product of a function of and a function of :

where we have used , without the label .

The power spectrum of the general solution is

Clearly, for a given moment of time , the spectrum is a modulated function of . For the modulation to take form of a strict periodic oscillation, the phase should be a linear function of . The oscillations vanish for traveling waves and have the maximal depth, up to the appearance of zeros, for standing waves. In principle, and could themselves be complicated functions of , but for the moment we do not consider this possibility.

For illustration, we show in Fig.1 a model spectrum () plotted for a discrete set of wavenumbers . The zeros in the spectrum, marked by blue stars, move and prolifirate in the course of time, in the sense that they gradually arise at new frequencies, and the distance between them decreases. The moving zeros and moving maxima will be inherited and fixed (possibly, with a phase shift) in the power spectrum at the stage after the transition.

Indeed, the general solution of Eq.(5) after the transition is . It is the coefficients that become oscillatory functions of . The moving features become fixed features at some particular wavenumbers, thus defining the “lucky” and “unlucky” oscillators. If the transition can be approximated as a sharp event occuring at some , then by joining the general solutions for the function and its first time derivative at , we find for the coefficient in the growing solution

Obviously, there are no final spectrum modulations if the incoming field consists of traveling waves ( or ), and the modulations have maximal depth if the waves are standing (). The relevant set of maxima is determined by the set of where the function has a maximum, starting from . The smallest and hence the largest spatial scale is expected to be the most pronounced observationally. For such long wavelengths, the metric perturbations cannot be generically neglected. Note that if the post-transition medium is CDM, then there must be oscillations in the CDM power spectrum.

One can see that it is only a very high degree of organization of the field before the transition, – standing waves with phases proportional to , – that can lead to the emergence of periodic Sakharov oscillations in the post-transition pressureless matter and in the associated metric perturbations.

The power spectra of cosmological fields in the recombination era determine the angular power spectrum of cosmic microwave background anisotropies observed today. Depending on whether the perturbations are realised as traveling or standing waves, the CMB spectra will be strongly different. This is best illustrated with the help of gravitational waves. In the case of gravitational waves only gravity is involved, so one should not worry about the “acoustic physics” and the role of various matter components. The decoupling of photons from baryons at the last scatering surface has no effect on gravitational waves themselves, but for the photons it is very important in which gravitational field they start their journey and propagate.

In Fig.2 we show two power spectra of gravitational waves given at and two corresponding CMB temperature spectra caused by them (more details in BG ). The red (wavy) line describes the physical spectrum formed by (quantum-mechanically generated) standing waves, whereas the grey (smooth) line shows the alternative background formed by traveling waves. The power spectrum of the alternative background was chosen to be an envelope of the physical one, so that the broad-band powers in the two spectra are approximately equal, except at very small ’s. The CMB spectra are placed right above the underlying gravitational wave spectra in order to demonstrate the almost one-to-one correspondence between their features in -space and -space. A similar correspondence holds for the power spectrum of first time-derivative of the field and CMB polarization spectra for which it is responsible BGP . It is important to note that the planned new sensitive measurements of CMB polarization and temperature (e.g. Spider ) may be capable of identifying the first cycle of oscillations in the physical gravitational-wave background.

## Iii Current observations of oscillations in power spectra of matter and CMB.

It should be clear from the discussion above that the Sakharov oscillations are not trivial acoustic waves in relativistic plasma. Such acoustic waves, expressing the variability of physical quantities in space and time, always exist, in the sense that they are general solution to the density fluctuation equation. The Sakharov oscillations are something much more subtle. They are the variability in the post-transition power spectrum, that is, oscillations in Fourier space. At late times, the oscillatory shape of the matter power spectrum remains fixed. The oscillations define the preferred wavenumbers and spatial scales, in agreement with the standing-wave pattern of the pre-transition field.

Oscillations in the final power spectrum do not arise simply as a result of a “snapshot” of oscillations in the baryon-photon fluid or as an “impression” of acoustic waves in the hot plasma of the early universe onto the matter distribution. And they are neither the result of the propagation of spherical sound waves up to the “sound horizon” before recombination, nor the result of “freezing out” of traveling sound waves at decoupling. The event when the plasma becomes transparent can make the Sakharov oscillations visible, but this is not the reason why they exist. Periodic structures in the final power spectrum arise only if the sound waves in relativistic plasma (as well as the associated metric perturbations) are standing waves with special phases. The oscillations in the power spectrum do not arise at all if the sound waves are propagating. It is also clear from the discussion above that the phenomenon of oscillations is not specific to baryons. The oscillations are present, for example, in the power spectrum of metric perturbations accompanying matter fluctuations and in gravitational waves.

It appears that actual observations have revealed convincing traces of Sakharov oscillations in the distribution of galaxies. Existing and planned surveys concentrate on the distribution of luminous matter (baryons) and therefore the spectral features are often called the baryon acoustic oscillations (BAO). The structures in the power spectrum are Fourier-related to the spikes in the two-point spatial correlation function. Both characteristics have been measured in galaxy surveys, (e.g. Cole05 , Eis05 , Perc07 , corrf3 ; the last citation contains many references to previous work.)

Of course, the ideal picture of standing waves in the early plasma is blurred by the multi-component nature of cosmic fluid and by the variety of astrophysical processes happenning on the way to the observed spatial distribution of nonrelativistic matter. This makes the oscillatory features much smoother and much more difficult to identify. Moreover, the measurement of our own particular realization of the inherently random field is only an estimate of the theoretical, statistically averaged, power spectrum, such as Eq.(4). Nevertheless, the impressive observations of recent years gave significant evidence of the existence of Sakharov oscillations.

A similar situation takes place in the study of CMB temperature and polarization. The difference between smooth and oscillatory underlying spectra for the ensuing CMB anisotropies was illustrated by gravitational waves in Fig.2. Density perturbations are more complicated because they include the individual power spectra of fluctuations in matter components, the velocity of the fluid which emits and scatters CMB photons (the velocity and the associated Doppler terms require careful definitions), and gravitational field perturbations. Surely, the observed peaks and dips in CMB temperature angular spectrum , now measured up to high multipoles ACT , are a reflection of oscillations in the underlying power spectra at the time of decoupling . (A link with the phenomenon of Sakharov oscillations, in some generalized sense, was mentioned in Jorg95 , BG .) It is very likely that the oscillations in at relatively high ’s are a direct reflection of standing-wave pattern of density variations in baryon-electron-photon plasma itself, so they are “acoustic” signatures. In contrast, the structures at the lowest ’s are probably having a considerable contribution from the pre-transition metric perturbations, which were inherited at the time of transition , mostly by the gravitationally dominant cold dark matter, so these structures are more like “gravitational” peaks and dips BG . [The current cosmological literature emphasizing the “acoustic” side of the problem incorrectly claims that there should not be oscillations in the power spectrum of CDM.]

It should be remembered, however, that the decomposition of the total CMB signal into different contributions is not unambiguous, and the interpretation may depend on coordinate system (gauge) chosen for the description of fluctuations. In the so-called Newtonian gauge the decomposition of the total signal is presented in Fig.3, taken from Chal04 . The dominating SW contribution (SW stands for Sachs-Wolfe) is a combination of variations of the metric and photon density.

We can make the following intermediate conclusions. First, for the Sakharov oscillations to appear in the final matter power spectrum, they must be encoded from the very beginning in the power spectrum of primordial cosmological perturbations, as a consequence of standing waves. Therefore, the Sakharov oscillations must have truly primordial origin (quantum-mechanical, as we argue below). Second, the very existence of periodic structures in the power spectra of matter and CMB is not a lesser revelation about the Universe than those future discoveries that will hopefully be made with the help of these “standard rulers”. In particular, in the case of data from galaxy surveys, it is important to be sure that we are dealing with manifestations of Sakharov oscillations, and not with something else. If they are Sakharov oscillations, then the phases were remembered for 13 billion years. Third, at some elementary level the Sakharov oscillations can be tested in laboratory conditions. This is a difference in fates of traveling and standing waves in a medium in which the sound speed changes from large values to zero. It would be useful to perform this experimental demonstration.

## Iv Quantum mechanics of the very early Universe.

It is appropriate to start this section with one of the last photographs of A.D.Sakharov (see Fig.4). It shows the intermission in the meeting chaired by Sakharov at which the present author (among other enthusiastic speakers) argued that if primordial cosmological perturbations were generated quantum-mechanically, then the result would be not just something, but very specific quantum states known as squeezed vacuum states, and why this should be important observationally. The notions of the vacuum, a squeezed vacuum and a displaced vacuum (coherent states) sounded suspicious to the audience, but Sakharov remained silent. At some crucial point he astonished me by the question “which variable specifically is squeezed ?”. Such a question can be asked only by someone who is perfectly well familiar with the discussed subject and deeply understands its implications.

Indeed, from the sketch in Fig.5 one can see that simple quantum states of a harmonic oscillator can greatly differ in mean values and variances of conjugate variables. For example, squeezed coherent states can be squeezed, i.e. have very small uncertainties, either in the number of quanta or in the phase. This leads to different observational results. I was glad to answer Sakharov’s question, because a squeezed vacuum state can be squeezed only in phase. The arising correlation of the and modes is equivalent to the generation of a standing wave (a two-mode squeezed vacuum state, more details are given in GrLH and GrWhel ). The appearance of the standing-wave pattern is not surprising if one thinks of the generating process as the creation of pairs of particles with equal energies and oppositely directed momenta. Moreover, the phase, almost free of uncertainties in strongly squeezed vacuum states, smoothly depends on , as the oscillators with different frequencies start free evolution (rotation of a higly squeezed ellipse in the plane) after the completion of the process of generation (squeezing of the vacuum circle into an ellipse). This provides the prerequisites for the future Sakharov oscillations.

The generation of excitations in physically different degrees of freedom – relic gravitational waves and primordial density perturbations – is described by essentially the same equations. The equation for gravitational-wave mode functions is

(6) |

while the equation for metric perturbations describing the density perturbation degree of freedom is

(7) |

where the variable is also known as a curvature perturbation. Surely, equations (6), (7) can also be written in the form of the “master equation”, Eq.(5). The function in Eq.(7) is not the constant that Sakharov Sakh65 uses in the equation of state, but for simple equations of state the scale factor is a power-law function and is then a constant. In this case, equations (6) and (7) are identically the same, and they have general solutions in terms of the Bessel functions.

The two-mode Hamiltonian

(8) |

is common for these two degrees of freedom, with the coupling function for gravitational waves and for density perturbations. The coupling functions coincide if . As a result of the Schrodinger evolution, the initial vacuum state of cosmological perturbations (ground state of the corresponding time-dependent Hamiltonian) evolves into a two-mode squeezed vacuum (multi-particle) state. In other words, cosmological perturbations are quantum-mechanically generated as standing waves GrLH , GrWhel .

The simplest models of the initial stage of expansion of the Universe are described by power-law scale factors . (The four cases of the initial stage considered by Sakharov Sakh65 also belong to this category.) Such gravitational pump fields generate gravitational waves (t) and density perturbations (s) with approximately power-law primordial spectra:

(9) |

where , and we are using Mpc. The amplitudes and are independent unknowns, but according to the theory based on Eqs. (6), (7) and (8) they should be of the same order of magnitude: , where is the Hubble parameter at the initial stage of expansion. [The inflation theory also uses the same superadiabatic (parametric) amplification mechanism, which was originally worked out for gravitational waves gr74 , GrWhel . However, after blind wanderings between variables and gauges, inflationists arrived at what they call the “standard”, or even “classic”, result of inflationary theory. Namely, the prediction of arbitrarily large in the limit of Harrison-Zeldovich-Peebles spectrum , and, moreover, for any strength of the generating gravitational field, i.e. for any value of the Hubble parameter of inflationary de Sitter expansion .] It is common to characterize the contribution of gravitational waves to the CMB by the ratio .

Our analysis ZG of the 7-year Wilkinson Microwave Anisotropy Probe data (WMAP7) has resulted in and as the respective maximum likelihood values in 3-parameter and marginalized 1-parameter searches. The uncertainties are still large, so these numbers can only be regarded as indications of a possible real signal. The relic gravitational waves are very difficult to register but they are the cleanest probe of the very early Universe gr74 , grLett76 , GrWhel . This is why they are in the centre of several programs aimed at their identification. The Sakharov oscillations are an element of the whole picture of quantum-mechanically generated cosmological perturbations, and hence the detection of relic gravitational waves would be a huge support for the entire theoretical framework.

## V Expected results of the ongoing observations. Conclusions.

The prospects of measuring relic gravitational waves with the help of data from the currently operating Planck mission appear to be good. In Fig.6, taken from ZG , we show the expected signal-to-noise ratio with which the signal will be observed assuming that the indications found in WMAP7 data are real. A big obstacle is the foreground contamination which should be carefuly dealt with. The ability, ranging from excellent to none, of removing contamination is parameterized by the parameter . We are also working with the pessimistic case, in which and the nominal instrumental noise in the polarization channel at each frequency is increased by a factor of 4. We see from the figure that the ratio can be as large as , and even in the pessimistic scenario it remains at the interesting level .

As was already mentioned above, the planned dedicated observations (e.g. Spider ) may even be able to outline the first cycle in the oscillatory power spectrum of the gravitational wave background.

In general, we can conclude that the originally proposed Sakharov oscillations, as well as related phenomena whose existence can be traced back to the earliest moments of our Universe, are right in the focus of current fundamental research.

## Acknowledgements

The author is gratful to Dr. Wen Zhao for help.

## References

- (1) A.D.Sakharov. Soviet Physics JETP, 22, 241 (1966) [Russian original: ZhETF, 49, 345 (1965)]
- (2) R.A.Sunyaev and Ya.B.Zeldovich. Astrophys. and Space Sci., 7, 3 (1970)
- (3) Ya.B.Zeldovich and I.D.Novikov. Relativistic Astrophysics. Vol.II (University Chicago Press, 1983)
- (4) P.J.E.Peebles and J.T.Yu. Astrophys. J. 162, 815 (1970)
- (5) P.J.E.Peebles. Letter of May 30, 1990
- (6) L.P.Grishchuk. In “General Relativity and John Archibald Wheeler”, Eds. I.Ciufolini and R.Matzner (Springer, New York, 2010) pp.151-199. [arXiv:0707.3319]
- (7) L.P.Grishchuk. arXiv:1012.0743
- (8) S.Bose and L.P.Grishchuk. Phys. Rev. D66, 043529 (2002)
- (9) D.Baskaran, L.P.Grishchuk, A.G.Polnarev, Phys. Rev. D74, 083008 (2006)
- (10) A.A.Fraisse et al. arXiv:1106.3087
- (11) S.Cole et al. Mon. Not. R. Astron. Soc., 362, 505 (2005)
- (12) D.J.Eisenstein et al. Astrophys. J., 633, 560 (2005)
- (13) W. Percival et al. Astrophys. J. 657, 645 (2007)
- (14) F.Beutler et al., arXiv:1106.3366
- (15) R.Hlozek et al., arXiv:1105.4887
- (16) H.E.Jorgensen, E.Kotok, P. Naselsky, I.Novikov, Astron. Astrophys. 294, 639 (1995)
- (17) A. Challinor, Lect. Notes in Physics, 653, 71 (Springer, 2004) [arXiv:astro-ph/0403344]
- (18) L.P.Grishchuk. In “Quantum Fluctuations”, Eds. S.Reynaud, E.Giacobino, and J.Zinn-Justin (Elsevier Science B.V., 1997) pp.541-561.
- (19) L.P.Grishchuk. Soviet Physics JETP, 40, 409 (1975) [Russian original: ZhETF, 67, 825 (1974)]
- (20) W.Zhao and L.P.Grishchuk. Phys. Rev. D82, 123008 (2010)
- (21) L. P. Grishchuk Pis’ma Zh. Eksp. Teor. Fiz. 23, 326 (1976) [Sov. Phys. JETP Lett. 23, 293 (1976); http://www.jetpletters.ac.ru/ps/1801/article_27514.pdf]