The constraints on the power spectrum of relic gravitational waves from current observations of large-scale structure of the Universe
Within the framework of cosmological inflationary models, relic gravitational waves arise as a natural consequence of the quantum nature of primordial space-time metric fluctuations and a validity of the general theory of relativity. Therefore, the detection of cosmological gravitational waves would...
Збережено в:
| Дата: | 2005 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Головна астрономічна обсерваторія НАН України
2005
|
| Назва видання: | Кинематика и физика небесных тел |
| Теми: | |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/79643 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | The constraints on the power spectrum of relic gravitational waves from current observations of large-scale structure of the Universe / B. Novosyadlyj, S. Apunevych // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 199-204. — Бібліогр.: 11 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-79643 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-796432015-04-04T03:01:38Z The constraints on the power spectrum of relic gravitational waves from current observations of large-scale structure of the Universe Novosyadlyj, B. Apunevych, S. MS3: Physics of Stars and Galaxies Within the framework of cosmological inflationary models, relic gravitational waves arise as a natural consequence of the quantum nature of primordial space-time metric fluctuations and a validity of the general theory of relativity. Therefore, the detection of cosmological gravitational waves would be the strongest evidence in support of basic assumptions of the inflation theory. Although the tracing gravitational waves by polarization patterns on last scattering surface of the cosmic microwave background is only planned for forthcoming experiments, some general constraints on the tensor mode of metric perturbations (i.e., gravitational waves) can be established right now. We present a determination of the amplitude of the relic gravitational waves power spectrum. An indirect best-fit technique was applied to compare observational data and theoretical predictions. As observations we have used data on the large-scale structure of the Universe and anisotropy of the cosmic microwave background temperature. The conventional inflationary model with 11 parameters has been investigated, all parameters were jointly evaluated. This approach gave us a possibility to find parameters of the power spectrum of gravitational waves along with statistical errors. 2005 Article The constraints on the power spectrum of relic gravitational waves from current observations of large-scale structure of the Universe / B. Novosyadlyj, S. Apunevych // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 199-204. — Бібліогр.: 11 назв. — англ. 0233-7665 http://dspace.nbuv.gov.ua/handle/123456789/79643 en Кинематика и физика небесных тел Головна астрономічна обсерваторія НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
MS3: Physics of Stars and Galaxies MS3: Physics of Stars and Galaxies |
| spellingShingle |
MS3: Physics of Stars and Galaxies MS3: Physics of Stars and Galaxies Novosyadlyj, B. Apunevych, S. The constraints on the power spectrum of relic gravitational waves from current observations of large-scale structure of the Universe Кинематика и физика небесных тел |
| description |
Within the framework of cosmological inflationary models, relic gravitational waves arise as a natural consequence of the quantum nature of primordial space-time metric fluctuations and a validity of the general theory of relativity. Therefore, the detection of cosmological gravitational waves would be the strongest evidence in support of basic assumptions of the inflation theory. Although the tracing gravitational waves by polarization patterns on last scattering surface of the cosmic microwave background is only planned for forthcoming experiments, some general constraints on the tensor mode of metric perturbations (i.e., gravitational waves) can be established right now. We present a determination of the amplitude of the relic gravitational waves power spectrum. An indirect best-fit technique was applied to compare observational data and theoretical predictions. As observations we have used data on the large-scale structure of the Universe and anisotropy of the cosmic microwave background temperature. The conventional inflationary model with 11 parameters has been investigated, all parameters were jointly evaluated. This approach gave us a possibility to find parameters of the power spectrum of gravitational waves along with statistical errors. |
| format |
Article |
| author |
Novosyadlyj, B. Apunevych, S. |
| author_facet |
Novosyadlyj, B. Apunevych, S. |
| author_sort |
Novosyadlyj, B. |
| title |
The constraints on the power spectrum of relic gravitational waves from current observations of large-scale structure of the Universe |
| title_short |
The constraints on the power spectrum of relic gravitational waves from current observations of large-scale structure of the Universe |
| title_full |
The constraints on the power spectrum of relic gravitational waves from current observations of large-scale structure of the Universe |
| title_fullStr |
The constraints on the power spectrum of relic gravitational waves from current observations of large-scale structure of the Universe |
| title_full_unstemmed |
The constraints on the power spectrum of relic gravitational waves from current observations of large-scale structure of the Universe |
| title_sort |
constraints on the power spectrum of relic gravitational waves from current observations of large-scale structure of the universe |
| publisher |
Головна астрономічна обсерваторія НАН України |
| publishDate |
2005 |
| topic_facet |
MS3: Physics of Stars and Galaxies |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/79643 |
| citation_txt |
The constraints on the power spectrum of relic gravitational waves from current observations of large-scale structure of the Universe / B. Novosyadlyj, S. Apunevych // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 199-204. — Бібліогр.: 11 назв. — англ. |
| series |
Кинематика и физика небесных тел |
| work_keys_str_mv |
AT novosyadlyjb theconstraintsonthepowerspectrumofrelicgravitationalwavesfromcurrentobservationsoflargescalestructureoftheuniverse AT apunevychs theconstraintsonthepowerspectrumofrelicgravitationalwavesfromcurrentobservationsoflargescalestructureoftheuniverse AT novosyadlyjb constraintsonthepowerspectrumofrelicgravitationalwavesfromcurrentobservationsoflargescalestructureoftheuniverse AT apunevychs constraintsonthepowerspectrumofrelicgravitationalwavesfromcurrentobservationsoflargescalestructureoftheuniverse |
| first_indexed |
2025-07-06T03:40:12Z |
| last_indexed |
2025-07-06T03:40:12Z |
| _version_ |
1836867348557987840 |
| fulltext |
THE CONSTRAINTS ON THE POWER SPECTRUM
OF RELIC GRAVITATIONAL WAVES FROM CURRENT OBSERVATIONS
OF LARGE-SCALE STRUCTURE OF THE UNIVERSE
B. Novosyadlyj, S. Apunevych
Astronomical Observatory, Ivan Franko National University of Lviv
8 Kyryla i Methodia Str., 79005 Lviv, Ukraine
e-mail: novos@astro.franko.lviv.ua, apus@astro.franko.lviv.ua
Within the framework of cosmological inflationary models, relic gravitational waves arise as a natu-
ral consequence of the quantum nature of primordial space-time metric fluctuations and a validity
of the general theory of relativity. Therefore, the detection of cosmological gravitational waves
would be the strongest evidence in support of basic assumptions of the inflation theory. Although
the tracing gravitational waves by polarization patterns on last scattering surface of the cosmic
microwave background is only planned for forthcoming experiments, some general constraints on
the tensor mode of metric perturbations (i.e., gravitational waves) can be established right now.
We present a determination of the amplitude of the relic gravitational waves power spectrum.
An indirect best-fit technique was applied to compare observational data and theoretical pre-
dictions. As observations we have used data on the large-scale structure of the Universe and
anisotropy of the cosmic microwave background temperature. The conventional inflationary model
with 11 parameters has been investigated, all parameters were jointly evaluated. This approach
gave us a possibility to find parameters of the power spectrum of gravitational waves along with
statistical errors.
INTRODUCTION
For the last decade, due to the progress in observations, the cosmology entered a new stage of its develop-
ment. This stage was signalized by a new generation of experiments aimed to measurements of anisotropies of
the cosmic microwave background (CMB). These were the balloon-borne BOOMERanG, MAXIMA, Archeops,
the ground-based interferometers DASI, CBI, VSA. And probably, the most important one is the successor of
the COBE space mission, Wilkinson Microwave Anisotropy Probe (WMAP) that published in 2003 the one-year
observation results. WMAP has carried out measurements of the CMB over the whole sky with an unprecedented
angular resolution and high sensitivity of detectors. The data and Web-links for these experiments are available
at the Web-site of the Legacy Archive for the Microwave Background Data Analysis (LAMBDA) project [5].
The CMB explorations were complemented with extensive studies of expansion dynamics of the Universe by
means of measuring distances to Supernovae and large-scale structure surveys.
Advances in the quality of experimental data manifestly call for a model capable to explain a whole set of
collected data. Now, it is well understood that the simplest cosmological models cannot match the observations
adequately, as, e.g., the standard flat CDM model with a scale-invariant power spectrum of density fluctua-
tions. The observations advance the complication of the model, the number of parameters increases as well as
the number of phenomena encompassed by the theory. Today, the elaborate inflationary models need about
11 parameters for the proper description of the reality.
Cosmological parameters could be classified as follows:
• The parameters related to a background model of a homogeneous and isotropic Universe. The evolution of
the Universe in this model is determined by the amount of energy densities of different components in ratio
to the critical one Ωi = ρi/ρcr. These are densities of the baryon component Ωb, hot dark matter (massive
neutrinos) Ων , cold dark matter Ωcdm, and the density parameter for dark energy (i.e., the cosmological
constant or quintessence) Ωde.
• The global properties of the Universe. According to the Friedmann equations, sum of density parameters
gives a unity for a spatially flat Universe, or 1 − Ωk for the curved one, so Ωk is a curvature parameter.
The Hubble constant H0 = 100 h km s−1 Mpc−1 is regarded as a global parameter too.
c© B. Novosyadlyj, S. Apunevych, 2004
199
• Parameters associated with the inflation. According to an inflationary scenario, the large-scale structure
of the Universe is assumed to be formed due to the growth of primordial matter density perturbations
because of the gravitational instability. The perturbations have an adiabatic nature and originate from
quantum fluctuations stretched to the cosmological scales during the stage of the exponential expan-
sion – the inflation. Perturbations are usually described by their power spectra in the most general form
Ps(k) = Ask
ns for scalar and Pt(k) = Atk
nt for tensor mode of perturbations.
• The list of parameters has some additional parameters, like the late re-ionization parameter τc and biasing
parameters to relate the distribution of density to the spatial distribution of real astrophysical objects,
e.g., galaxies.
This parameter set is required to get the model predictions which are to be compared with the observations
in order to find out how good the theoretical assumptions match the results of current observations. As far
as these predictions depend on the parameters quite nonlinearly, the determination of the best-fit value for
one parameter requests the determination of a full set of parameters, by means of statistics. Actually that is
the main task of this investigation, sometimes referred as a testing of cosmological models.
In this paper, we shall pay attention to one parameter among many others, namely, the primordial power
spectrum of tensor mode of space-time metric perturbations (relic gravitational waves). We shall use the most
common kind of cosmological model and analyse some particular inflation models. Sometimes, this model is
designated as a concordance model, or even standard model. The purpose of this model is to give a plausible
explanation for the large-scale structure of the Universe and its global properties.
THE NATURE OF RELIC GWs
Indeed, gravitational waves are very simple conceptually but proved to be extremely elusive for the detection.
Gravitational waves ought to be emitted by any physical system with changes in the quadrupole distribution
of the stress-energy density. This follows from equations of general relativity as a free solution (wave solution)
for small linear space-time metric perturbations on the background metric.
Obviously, only astrophysical objects could produce gravitational waves of significant amplitudes. The num-
ber of probable astrophysical sources is under discussion for observational programmes (see for review [4]).
There are by now no direct successful detections of any kind of sources. For cosmological GWs, in fact, a very
small chance exists to be detected by any human-made antenna because of their super-long wavelengths.
The cosmological GWs play the role of the most ancient relic in our Universe. They have their origin in
the first moments of the Universe evolution, presumably the times of inflation. If we assume the general theory
of relativity to be valid in those times and quantum zero-point oscillations to exist, so we come to conclusion
about the inescapable existence of GWs. A variable gravitational field in a very early Universe parametrically
amplifies quantum oscillations, making up a stochastic gravitational background. The amplitude of these GWs
is determined by energy scales at the moment of emission, so for inflation this amplitude determines the energy
scale of processes that give a rise for inflation. So, the nature of relic gravitational waves is very fundamental
for physics. In fact, both scalar mode and tensor modes of space-time metric perturbations share a common
origin from a single physical process – quantum oscillations.
The subsequent evolution of metric linear perturbations is completely described by the gauge-invariant
formalism for cosmological perturbations. The relic GWs are represented by tensor mode of space-time metric
perturbations on the isotropic and homogeneous expanding background. As far as GWs do not interact with
the rest of the medium in an expanding Universe, their amplitude should decrease in course of time (see [1] for
an exhaustive review).
The gauge-invariant theory leads to the equation for the evolution of tensor perturbations of space-time
metrics in vacuum or a perfect fluid:
Ḧ(T ) + 2
ȧ
a
Ḣ(T ) + (2K + k2)H(T ) = 0, (1)
where H(T ) is the gauge-invariant amplitude of tensor perturbations for two polarizations, a is a scale factor,
K = (−1, 0, 1) is the curvature index, and k is the wavenumber. The dots denote the derivatives with respect
to the conformal time η. The solution of this equation is a propagating damped wave.
OUTLINE OF THE METHOD
As it was stated above, both tensor and scalar modes of space-time metric perturbations have the same origin.
The perturbations of scalar mode are connected to the density perturbations so they eventually lead to the for-
mation of galaxies, clusters, and voids, i.e., the large-scale structure of the Universe. Since tensor perturbations
200
do not lead to the formation of the large-scale structure, cosmological GWs can not be revealed by the present
state of the observable structure. Both scalar and tensor perturbations produce the power in the CMB angular
power spectrum due to the Sachs–Wolfe effect. Of course, because of its specific polarization properties, the relic
GWs should generate a particular polarization pattern of CMB anisotropies, but the detection of polarization
patterns in CMB is the matter of the future.
10 100 1000
l
0
1e-10
2e-10
3e-10
4e-10
5e-10
6e-10
l(
l+
1)
C
l /2
π,
μ
Κ
2
Scalar
Tensor
CMB Power Spectrum
10 20 30 40 50
0
1e-10
2e-10
T/S=1.0, ΩΛ=0.7, Ω
b
=0.05, Ω
CDM
= 0.25, n
s
= 1.0, n
T
= -0.5
Figure 1. The scalar and tensor contributions to the CMB angular power spectrum for a particular model
Therefore, we propose an indirect method. As far as the CMB power spectrum consists of contributions
from scalar and tensor modes, we can extract the last one if we precisely know the scalar contribution from
the large-scale structure. Also, the relic GWs have extra-long wavelengths of a particle horizon size, so the CMB
power spectrum from them should manifest a fast decrease with higher multipole moments (smaller scales).
On the contrary, the density perturbations at larger multipoles produce a feature-rich CMB power spectrum,
a series of acoustic peaks. Thus, the physical properties of tensor mode of cosmological perturbations of
space-time metric can be estimated on the basis of observational data on an angular power spectrum of CMB
combined and the large-scale structure of the Universe in a wide range of scales – from the galactic ones to
the size of a particle horizon, from 10−3 to 104 Mpc. This way, we can constrain a total level of the intensity of
gravitational waves. In the multipoles range of � ∼ 10 – 20, the scalar and tensor contributions are comparable
(Fig. 1). So, the precision of the results is to be determined by the precision of datapoints within this range.
This method is basically a normalization of the angular power spectrum.
THE BUILDING PREDICTIONS
Unfortunately, the cosmological models are quite complicated and the most of predictions could not be reduced
to analytical functions with cosmological parameters as arguments. Therefore, for the given parameter set,
one has to run numerical computations to make predictions. One way is the “brute-force” method, when
the numerical packages like CMBfast [6] are utilized. Since the testing requires a large number of models to
be evaluated, so even quite fast codes need a huge computational resources to accomplish the task. Another
demerit lies in the so-called “black box” uncertainty of numerical calculations, when the physical processes are
hidden inside computations. So, we propose semi-analytical approaches to make predictions of the CMB power
spectrum.
To obtain a tensor contribution to the CMB power spectrum within a low multipoles range (2 < � < 20),
we have developed an analytical approximation able to reproduce a computed spectrum with a high accuracy,
for a wide range of parameters values. This approximation has the form:
�(� + 1) CT
� =
A(nt, Ω0)
l + b(nt, Ω0)
· K�(Ωde) × exp(−C(nt, Ω0) · �2 + D(nt, Ω0) · �), (2)
201
where the coefficients A, b, C, and D represent polynomial functions of nt and Ω0 = 1 − Ωk, K�(Ωde) is
the amplification factor for the CMB power spectrum caused by dark energy. Analytical formulae for them are
presented in our paper [7].
A semi-analytical approximation for the scalar contribution to the CMB power spectrum for the multipoles
range 2 < � < 20 was developed in our previous paper [3] and is used here. This approximation combines
the speed of computations and a clear unobscured physical meaning of processes that cause the anisotropies in
CMB. The angular power spectrum of CMB at higher multipoles has a number of peaks separated with dips
and manifests the damping of an average amplitude to the highest �. This features are explained by acoustic
oscillations in a photon–baryon fluid set up by adiabatic perturbations at the entering the sound horizon. Instead
of calculations of power for each � within a wide range, we propose to describe the whole shape of the CMB
power spectrum by positions and amplitudes of the peaks and dips. The insignificant loss of information will
be a price for the computations speed-up. The number of analytical approximations was developed for heights
and positions of first three peaks and for position of the first dip in the CMB power spectrum in the models
considered, see [3].
OBSERVATIONAL DATA
Of course, the results of testing of the model strongly depend on the data used. The next important requirement
the data should meet is the statistical independence, i.e., the covariance matrix of datapoints errors should be
diagonal. Here we have used the same set of observational data as in [2, 3], with some changes. The amplitudes
and positions of the 1st and 2nd acoustic peaks and position of the first dip have been taken from the results of
the WMAP mission team [8], position and height of the third peak have been taken from the last recompilation
of the results of the BOOMERanG experiment [10]. At large angular scales (a lower multipoles range), we use
all datapoints from the COBE experiment and WMAP [5], except dipole.
The CMB data are complemented by the large-scale structure data: mass function and spatial distribution of
rich galaxy clusters, temperature function of X-ray clusters, Ly-α forests of absorption lines in distant quasars
spectra, peculiar velocities of galaxies. These LSS data can establish the amplitude and shape of a spatial
power spectrum of matter density perturbations. During the determination of parameters we have the problem
with the degeneracy in dependence of observational manifestations upon the parameters. This problem is
partially eliminated when we add additional measurements to the observational set: determinations of expansion
dynamics due to angular distances to Supernovae of Ia type, independent determinations of the Hubble constant,
constraints on the baryon content from the Big Bang nucleosynthesis theory.
There are in all 41 values in the list of observational datapoints along with 1σ statistical errors of their
measurements. We consider all measurements to be independent and take their probability distribution as
a normal distribution. These values and errors are described in details in the cited papers. Thus, if we compare
these results with those from [3], we obtain the answer to the question how improvements in measurements of
CMB anisotropies affect the determination of parameters of the observable Universe.
LIKELIHOOD ANALYSIS
Let us have N observational characteristics and we are searching the best-fit values of n cosmological parameters.
In other words, we have the parameter set:
−→
P = (ΩCDM , Ωde, Ων , Nν , Ωb, h, As, ns, At, nt, τc)
and the set of observations:
−→
D = (Ap1 , �p1 , Ap2 , �p2 , Ap3 , �p3 , �d1 , Cllow
, LSS, NS, h, SN).
To find the best-fit values for 11 parameters we are using the Levenberg–Marquardt algorithm [9] to minimize
the function
χ2 =
N∑
j=1
(
ỹj − yj
Δỹj
)2
, (3)
where ỹj is the observational value for some j-th characteristic, yj is its theoretically predicted value, Δỹj is
a statistical uncertainty for the measured value. Since the number of neutrino species Nν is a discrete value,
so we were searching for values of 10 parameters at Nν held fixed at 1, 2, 3. As in the previous papers,
instead of dimensional values of the amplitude of scalar mode power spectra As, we have used the dimension-
less value δh, defined as rms of fluctuation of the matter density on scale of the present horizon of particle,
As = 2π2δ2
h(3000 Mpc/h)3+ns .
202
To find errors for parameters values one has to carry out an exploration of the likelihood function profile
L ∝ e−
1
2 χ2
in a parametric space to determine confidential ranges (marginalizing). The estimations of con-
fidential ranges for parameters are mostly based on the direct likelihood function integration [9], a kind of
cumbersome computations. In [2, 3] we have proposed “economical” methods for the marginalization proce-
dure to avoid the direct integration. Here we propose a combined approach: the likelihood function profile for
parameter xk is built using the minimization of χ2 in a subspace of n− 1 parameters
L(xk) = e−
1
2 [χ2(xbf
i�=k,xk)−χ2
min] , (4)
where xbf
i�=k are the best-fit values of cosmological parameters (i = 1, 2, . . . , 12, i �= k), for which χ2-function
has the minimum at a fixed value of parameter xk. This kind of representation for L(xk) can be obtained from
the general integral form. A numerical experiment proves that the results from the proposed function L(xk) for
a confidential level estimation virtually coincide with the values obtained by integration.
RESULTS
For historical reasons, the common practice is to give a description of relic gravitational waves by the ratio of
amplitudes of tensor T and scalar contributions S to the quadrupole (� = 2) component of the CMB power
spectrum. However, probability distributions for At and T/S differ, and L(At) function is closer to the Gaussian
shape than L(T/S). So, we are using L(At) to determine upper bounds and to recalculate it then to T/S.
We define the upper bound A2σ
t at a confidential level 2σ as a value for which the square under the curve L(At)
has 95.4% of the total square under this curve from 0 to ∞.
Figure 2. Likelihood functions L(At) = exp[− 1
2
χ(At)] for various sets of observational data
In terms of our technique, CMB and LSS data have an insufficient statistical weight in order to provide
a simultaneous determination of both amplitude and slope of the tensor power spectrum. In other words, if we
would leave both At and nt as free parameters, then At could take an arbitrary large value under condition
nt → −∞. If we keep the lower bound for nt fixed, then the upper bound for At depends on it. This problem
finds its natural explanation by limitations of the indirect method used. We have no data to distinguish
the amplitude and slope separately. Fortunately, the majority of inflation models relate the nt with the slope
of the scalar power spectrum ns. So, we have also analysed the likelihood functions for the same observational
data and some generic inflation models: with a flat spectrum of tensor mode (nt = 0), natural inflation with
nt = ns − 1, and chaotic inflation with nt = 0.5(ns − 1). The upper 2σ constraints for them are the following:
A2σ
t = 1.9 · 10−5 for the first model and A2σ
t = 1.0 · 10−5 for the rest. The corresponding values for them
are T/S = 0.6 and 0.18. These three models are more interesting from the standpoint of manifestations of
tensor mode in the data of observations, so further analysis will include merely them. As far as the likelihood
functions L(At) for them are very similar, it is quite enough to analyse one of them, namely, we take a model
with nt = 0.5(ns − 1).
Figure 2 illustrates how observational data influence on the half-width of a likelihood function. As we
can see, the adding the observational data with the constraints on cosmological parameters themselves, like
the Hubble constant, Big Bang Nucleosynthesis, SNIa observations, and data on the large-scale structure,
203
decreases the confidence level for the models with high tensor amplitudes. At the confidential level of 2σ (95.4%),
this amplitude cannot exceed ∼ 20% of the scalar mode amplitude. In our previous estimations (see [3]), based
on the data of the balloon experiment BOOMERanG, this limitation was almost four times larger (at 1σ C.L.
the ratio was estimated as T/S = 1.7 for the model with free nt). So, we clearly see the achievements of WMAP
with high precision and covering over all sky.
The results for a whole parameter set are summarized in Table 1. The constraints are tabulated here for
the case of a chaotic inflation. As we can see, the accordance of these best-fit values with previous determina-
tions [2, 3] is quite satisfactory. The new precise measurements of CMB temperature anisotropies significantly
tightened confidential ranges and lowered upper constraints for Ων , τc and T/S. The results obtained agree
with determinations of other authors which used the WMAP data [11] (different definitions).
Table 1. Best-fit values of parameters, lower and upper bounds at 2σ (95.4%) confidential level
Ωde Ωm Ων Ωb h δh ns T/S τc
Best-fit 0.61 0.41 0 0.062 0.61 4.2 · 10−5 0.92 0 0
Lower bound 0.52 0.31 0 0.046 0.52 3.6 · 10−5 0.89 0 0
Upper bound 0.69 0.51 0.03 0.078 0.71 5.2 · 10−5 0.98 0.6 0.15
CONCLUSIONS
From the standpoint of statistics an interpretation of available observational data on large-scale structure of
the Universe and CMB does not require the presence of a considerable amount of relic gravitational waves.
According to the basic assumptions of the early Universe physics, these relic waves have an inescapable nature
and common origin with matter density fluctuations seeding a large-scale structure. Within the framework
of the inflation theory, the manifestations of the gravitational background can be quite prominent, as long as
the amplitude of relic GWs directly connected to energy scales of inflation, e.g., the Grand Unification Theory
energy scale. Thus, the upper bounds on the amplitude of GWs appear to be utmost important to provide
constraints on the moment and energy scales of the inflation allowing to discriminate among models of inflation.
The advances in measurements of the CMB in the space experiment WMAP substantially lowered the up-
per bound for the amplitude of tensor mode of perturbations (i.e., relic gravitational waves) to the level of
T/S ≤ 0.6 (95.4% C.L.) for models with the free slope parameter nt. For models with a flat power spectrum of
gravitational waves (nt = 0), or some close to that spectra (nt ∼ 1 − ns), this limit appears to be even lower,
T/S ≤ 0.18 (95.4%).
[1] Durrer R. The theory of CMB anisotropies // J. Phys. Studies.–2001.–5, N 2.–P. 177–215.
[2] Durrer R., Novosyadlyj B. Cosmological parameters from complementary observations of the Universe // Mon.
Notic. Roy. Astron. Soc.–2001.-324.–P. 560–572.
[3] Durrer R., Novosyadlyj B., Apunevych S. Acoustic Peaks and Dips in the Cosmic Microwave Background Power
Spectrum: Observational Data and Cosmological Constraints // Astrophys. J.–2003.–583, N 1.–P. 33–48.
[4] Grishchuk L. P. et al. Gravitational wave astronomy: in anticipation of first sources to be detected // Usp. Fiz.
Nauk.–2001.–171.–P. 3–59.
[5] [http://lambda.gsfc.nasa.gov/].
[6] [http://www.cmbfast.org].
[7] Novosyadlyj B., Apunevych S. Cosmological constraints on the amplitude of relic gravitational waves // J. Phys.
Studies.–2005.
[8] Page L. et al. First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Interpretation of the
TT and TE Angular Power Spectrum Peaks // Astrophys. J. Suppl. Ser.–2003.–148.–P. 233–241.
[9] Press, W. H. et al. Numerical recipes in FORTRAN.–New York: Cambridge Univ. Press., 1992.
[10] Ruhl J. E. et al. Improved Measurement of the Angular Power Spectrum of Temperature Anisotropy in the Cosmic
Microwave Background from Two New Analyses of BOOMERANG Observations // Astrophys. J.–2003.–599.–
P. 786–805.
[11] Spergel D. N. et al. First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Determination of
Cosmological Parameters // Astrophys. J. Suppl. Ser.–2003.–148.–P. 175–194.
204
|