# Constraints on alternative theories of gravity and cosmology

Constraints on alternative theories of gravity and cosmology

Constraints on alternative theories of gravity and cosmology

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**on**straints**on** **alternative** **theories** **of** **gravity** **and** **cosmology**Alex**and**er F. Zakharov 1,2 , Victor N. Pervushin 2 , Francesco De Paolis 3 ,Gabriele Ingrosso 3 **and** Achille A. Nucita 41 Institute **of** Theoretical **and** Experimental Physics,B. Cheremushkinskaya, 25, 117259, Moscow, Russia,email: zakharov@itep.ru2 Bogoliubov Laboratory for Theoretical Physics,Joint Institute for Nuclear Research, 141980 Dubna, Russia3 Department **of** Physics **and** INFN,University **of** Salento, CP 193, I-73100, Lecce, Italy4 XMM-Newt**on** Science Operati**on**s Centre, ESAC, ESA,PO Box 50727, 28080 Madrid, SpainAbstractOne could call 2006 as the year **of** **cosmology** since in the year two US scientists were awarded bythe Nobel prize for their studies **of** Cosmic Microwave Background (CMB) spectrum **and** anisotropy.Studies **of** CMB anisotropy d**on**e with the Soviet spacecraft Prognoz-9 by the Relikt-1 team arereminded. Problems **of** modern **cosmology** are outlined. We discuss c**on**formal **cosmology** parametersfrom supernovae data in brief. Two approaches to solve the basic problems **of** **cosmology**,such as dark matter **and** dark energy, are discussed, the first (st**and**ard) possibility is to introducenew particles, fields etc, the sec**on**d possibility is to try to change a **gravity** law to fit observati**on**aldata. We discuss advantages **and** disadvantages **of** the sec**on**d choice.Keywords: General Relativity **and** Gravitati**on**; Cosmology; Observati**on**al Cosmology; Cosmologicaltests; Supernovae; Cosmic Microwave Background Radiati**on**.1 Introducti**on**For a scientific community 2005 was the World year **of** physics due to publicati**on**s **of** the famousEinstein’s papers **and** the birth **of** modern (c**on**temporary) physics in 1905. The next year (2006) wasalso remarkable for a physical community (it was the year **of** astrophysics **and** **cosmology**) since in 2006the Nobel prize for CMB studies was presented to J. Mather **and** G. Smoot, moreover J. Mather **and**the COBE (COsmic Background Explorer) team was awarded by the Peter Gruber prize **on** **cosmology**in the same year at the General Assembly **of** the Internati**on**al Astr**on**omical Uni**on** in Prague. Nodoubt, 2007 will be recognized as the very important year for astrophysical (or probably for scientificin general) community since a number **of** **of** discoveries were d**on**e even at present date. For Russianspace science 2007 is the jubilee year since the first artificial satellite (Sputnik) was launched fifty yearsago **on** October 4, 1957.A structure **of** the paper is the following. In secti**on** 2 we remind results **of** CMB anisotropystudies d**on**e with the Soviet Relikt-1 missi**on**. In secti**on** 3 we outline a current status **of** cosmologicalstudies **and** parameters **of** c**on**formal cosmological models in brief. In secti**on** 4 we discuss a success **of****alternative** way to solve DM **and** DE problem (so-called, a geometrical approach or f(R)-models), wherea **gravity** law is substituting with changing classical Lagrangian for **gravity** **and** severe c**on**straints **on**parameters **of** so-called R n theory from Solar system data will be given. C**on**clusi**on**s will be presentedin secti**on** 5.2 CMB anisotropy studies, Relikt-1 & COBECosmic microwave background (CMB) existence was predicted in the framework **of** the the so called BigBang cosmological model [1, 2, 3, 4], however, first estimates CMB temperature T CMB ∼ 50 0 K were1

ather rough since it was assumed that the age **of** the Universe is about 2–3×10 9 years (because **of** theHubble c**on**stant estimate was very high H = 500 km/(s·Mpc) due to errors in distance measurementsat the time), but later Gamow re-estimated the temperature T CMB ∼ 6 0 K [5]. The CMB radiati**on**was discovered by A. Penzias **and** R. Wils**on** [6] in an unexpected way (**and** they awarded by the Nobelprize for the discovery in 1978). The physical meaning the discovery was explained in the introductorypaper [7] published in the same issue **of** the Astrophysical Journal as the paper [6] with a descripti**on****of** the discovery.Because the Earth moves in respect to the CMB, a dipole temperature anisotropy **of** the level **of** isexpected. A dipole temperature anisotropy **of** the level **of** ∆T/T = 10 −3 was observed, it corresp**on**ds toa peculiar velocity 380 km/s **of** the Earth towards the c**on**stellati**on** Virgo (first measurements **of** dipoleanisotropy were not very precise [8], but later the accuracy was significantly improved [10, 11, 12]).Moreover, based **on** results **of** ballo**on** measurements it was claimed several times about features **of**quadrupole anisotropy [12, 13], however, the quadrupole term is too high in these experiments asfurther studies had indicated, or reversely the realistic level **of** quadrupole anisotropy is to low to bedetected in the ballo**on** experiments.In 1983, in the Soviet Uni**on** the Relikt-1 experiment 1 was c**on**ducted aboard the Prognoz-9 spacecraft2 in order to investigate CMB radiati**on** from space for the first time in history. As many otherPrognoz missi**on**s, the scientific payload was prepared by the Space Research Institute **of** the SovietAcademy **of** Sciences. Dr. Igor Strukov was the principal investigator.The spacecraft Prognoz-9, had an 8 mm b**and** radiometer with an extremely high sensitivity **of**35 µK per sec**on**d **and** it was launched into a high apogee orbit with a 400,000 km semi-axis. The highorbit was a great advantage **of** the missi**on** since it allows to reduce **of** geomagnetic field impact **on**measurements. A disadvantage **of** the experiment was that the observati**on**s were c**on**ducted **on**ly in**on**e spectral b**and**, therefore, it was a freedom for a theoretical interpretati**on** **of** the results, in c**on**trast,multi-b**and** measurements provide very small room for **alternative** explanati**on**s **of** anisotropy.The radiometer scanned the entire celestial sphere for six m**on**ths. Computer facilities (**and** therefore,data analyzing) were relatively slow in the time. Preliminary analysis **of** anisotropy studiesindicate upper limits **on** anisotropy [15], but in this case even the negative results (upper limits) wereextremely important to evaluate a sensitivity to design detectors for next missi**on**s (including COBE).In 1986, at the Space Research Institute it was decided to prepare a next generati**on** **of** spaceexperiments to study the anisotropy **of** CMB, **and** start to develop the Relikt-2 project. The sensitivity**of** the detectors was planned to be in 20 times better than the Relikt-1 sensitivity. The Libris satellitewas scheduled to carry the Relikt-2 payload **and** the spacecraft was planned to be located near theLagrangian point L 2 (in the Sun – Earth system). 3 Originally, it was a plan to launch the Librisspacecraft in 1993–1994, however, the project has not been realized, basically due to a lack **of** funds.To prepare Relikt-2, the team members re-analyzed Relikt-1 data **and** finally in beginning **of** 1991 theydiscovered signatures **of** the quadrupole anisotropy, but I. Strukov required to check the c**on**clusi**on**sagain **and** again. The discovery **of** anisotropy by the Relikt-1 spacecraft was first reported **of**ficially inJanuary 1992 at the Moscow astrophysical seminar **and** the Relikt-1 team submitted papers in SovietAstr**on**omy Letters [16] **and** M**on**thly Notices **of** Royal Astr**on**omical Society [17] (so**on** after the paperswere published). Relikt-1 results are described in an adequate way at the at the **of**ficial NASA web-sitehttp://lambda.gsfc.nasa.gov/product/relikt/ 4 ”... The Relikt Experiment Prognoz 9, launched **on** 1July 1983 into a high-apogee (700,000 km) orbit, included the Relikt-1 experiment to investigate theanisotropy **of** the CMB at 37 GHz, using a Dicke-type modulati**on** radiometer. During 1983 **and** 1984some 15 milli**on** individual measurements were made (with 10% near the galactic plane providing some5000 measurements per point). The entire sky was observed in 6 m**on**ths. The angular resoluti**on** was5.5 degrees, with a temperature resoluti**on** **of** 0.6 mK. The galactic microwave flux was measured **and**the CMB dipole observed. A quadrupole moment was found between 17 **and** 95 microKelvin rms, with1 In Russian literature CMB radiati**on** is called as a relic radiati**on**, therefore the name Relic (”Relikt” in Russian)was chosen.2 In Russian ”Prognoz” means ”forecast”. Prognoz spacecrafts were designed originally for studies **of** Solar activity(Solar flares in particular) **and** its impact **on** geophysics, geomagnetic fields **and** space weather [14]. The spacecrafts werelaunched at high elliptical orbits with apogee about 200 000 km.3 The trajectories near the L 2 point are suitable for different space experiments, for example, the orbit was selectedfor the WMAP spacecraft (it is **on**e **of** the most successful missi**on** to study the CMB anisotropy **and** such an orbit wasplanned to be used for the future Millimetr**on** cryogenic telescope (it scheduled to be launched around 2016)).4 See, also, an article ”Cosmic microwave background radiati**on**” in Wikipedia, the free encyclopedia(http://en.wikipedia.org/wiki/Main Page).2

90% c**on**fidence level. A map **of** most **of** the sky at 37 GHz is available...” (also, references [16, 17],given at the NASA web-site).The Nobel Prize in Physics for 2006 was awarded to John Mather **and** George Smoot. The RoyalSwedish Academy **of** Sciences had issued the Press Release, dated **on** 3 October 2006, **on** The NobelPrize in Physics 2006. 5 The Royal Swedish Academy **of** Sciences has decided to award the Nobel Prizein Physics for 2006 jointly to John C. Mather (NASA Goddard Space Flight Center, Greenbelt, MD,USA), **and** George F. Smoot (University **of** California, Berkeley, CA, USA) ”for their discovery **of** theblackbody form **and** anisotropy **of** the cosmic microwave background radiati**on**”. Really, J. Mather wasthe project scientist **of** the COBE missi**on** **and** was resp**on**sible for measurements **of** CMB spectrum (hewas the PI **of** the Far InfraRed Absolute Spectrometer (FIRAS) experiment) **and** measurements **of** CMBspectrum were rather successful. G. Smoot did a great c**on**tributi**on** into measurements **of** the dipoleanisotropy [11] **and** the quadrupole **on**e [18], since he was PI **of** the differential microwave radiometer(DMR) aboard the COBE satellite **and** found the quadrupole anisotropy with these facilities, but bothanisotropies were discovered earlier by other people.The discovery **of** anisotropy by the Relikt-1 spacecraft was first reported **of**ficially in January 1992at the Moscow astrophysical seminar. Relikt-1 team submitted their paper in Soviet Astr**on**omy Letters**and** M**on**thly Notices **of** Royal Astr**on**omical Society **on** January 19, 1992 **and** **on** February 3, 1992,respectively.On April 21, 1992, G. Smoot (the head **of** DMR experiment aboard the COBE missi**on**) **and** his coauthorssubmitted a paper in Astrophysical Journal (Letters) [18]. On April 22, 1992, Smoot reportedat a press c**on**ference about the discovery **of** the CMB anisotropy with the COBE satellite. After thatmass media reported this results as the main science success. In 1992, COBE results about discovery**of** the CMB anisotropy were reported elsewhere. However, no doubt, the COBE collaborati**on** knewresults **of** Relikt-1 team **and** even quoted their upper limit **on** the quadrupole anisotropy published atthe paper.[15] Sometimes, results **of** both (COBE **and** Relikt-1) collaborati**on**s were presented at thesame scientific meetings. In particular, in June 1992, at the c**on**ference **on** Particle Astrophysics in Blois(France) results **on** discoveries **of** CMB anisotropy were presented at a plenary sessi**on** by representatives**of** both teams (Relikt-1 **and** COBE), namely M. V. Sazhin **and** G. Smoot. The Smoot’s paper waspublished in the c**on**ference proceedings [19], but unfortunately, eventually, Sazhin’s c**on**tributi**on** wasnot published there.However, summarizing, **on**e could say that since papers **of** the Relikt-1 team were submitted **on**January 19, 1992 **and** February 3, 1992 in Soviet Astr**on**omy **and** M**on**thly Notices **of** Royal Astr**on**omySociety respectively, but COBE paper was submitted **on** April 21, 1992, **on**e would c**on**clude that thediscovery **on**e quadrupole anisotropy was d**on**e by the Relikt-1 team **and** published in papers [16, 17].3 Precise **cosmology** & C**on**formal Cosmological ModelsSince 1998 people claimed that **cosmology** started to be a precise science. Really, if the st**and**ard cosmologicalparadigm was chosen, now we can evaluate cosmological with a precisi**on** about 10 % (typically**on**ly statistical errors are indicated **and** it is assumed that systematical errors are negligible). However,even in the framework **of** the st**and**ard paradigm **of** the Big Bang **cosmology**, our underst**and**ing thebest fit for cosmological model is changing. For example, in the beginning **of** 1998 it was found thatthe Universe is opened **and** Ω m ∼ 0.2 − 0.4 (based **on** observati**on**s **of** galactic clusters [20] (immediately,mass media had reported that eventually scientists proved that the Universe would be exp**and**ingforever), but at the end **of** 1998, it was c**on**cluded that the Universe is flat Ω m = 0.3, Ω Λ = 0.7 [21, 22](assuming that the SNeIa are st**and**ard c**and**les).We will remind basic relati**on**s for CC model parameters (see papers [23, 24] for details) c**on**sideringthe General Relativity with an additi**on**al scalar field Q, as usually people did to introduce quinessence[25, 26, 27, 28] (earlier, the approach was used for inflati**on**ary **cosmology**, see for example, paper [29]**and** references therein)∫S CR =d 4 x √ [−g − 1 6 R(g) + 1 ]2 ∂ µQ∂ µ Q − V (Q) , (1)5 Informati**on** from the Nobel prize committee web-site http://www.kva.se.3

where we used the natural units√3M Planck = = c = 1, (2)8πtherefore, we have the following expressi**on**s for density **and** pressure **of** the scalar field, p Q **and** ρ Q ,respectively [25, 26, 27, 28],p Q (t) = 1 2 ˙Q 2 + V (Q),**and** equati**on** **of** state (EOS) such as p Q = w Q ρ Q , whereρ Q (t) = 1 2 ˙Q 2 − V (Q), (3)w Q =12 ˙Q 2 − V (Q)12 ˙Q 2 + V (Q) , (4)(−1 ≤ w Q ≤ 1, for ”natural” potentials V (Q) ≥ 0). In c**on**trast with quintessence model where **on**e usestypically ˙Q 2 ≪ V (Q), below for CC model we will use an approximati**on** ˙Q 2 ≫ V (Q) (for a st**and**ardrepresentati**on** **of** the potential V (Q) = 1 2 mQ2 , where m is a mass **of** the field, the approximati**on**corresp**on**ds to a massless field model) **and** we havew Q =1 ˙Q 2212˙Q2= 1, (5)or **on** the other words, a rigid EOS for the scalar field p Q = ρ Q (p rig = ρ rig , since for our future needsan origin **of** the EOS is not important, hereafter, we will call the comp**on**ent such as the rigid matter).We suppose that a measurable interval is defined by the Weyl-like ratiods 2 meas = ds2 Einsteinds 2 units=g µν dx µ dx ν[g µν dx µ dx ν ] units(6)**of** two Einstein intervalsds 2 Einstein = g µν dx µ dx ν (7)**on**e **of** which ds 2 units plays a role **of** measurement units determined by **on**e **of** masses. The Weyl definiti**on**means that there is no device measuring the absolute value **of** any length. Nevertheless, this definiti**on**gives us a possibility to fit also the units **of** observati**on**s as two choices **of** observable variables. Thesechoices corresp**on**d to two different cosmological models: the St**and**ard Cosmology with observablemasses m sc = m **and** Einstein’s intervals in homogeneous approximati**on****and** the C**on**formal Cosmology, where all massesare scaled by the factor a **and** measurable intervalsds 2 sc = dt 2 − a 2 (dx k ) 2 = a 2 (η)[(dη) 2 − (dx k ) 2 ] (8)do not depend **on** the cosmological scale factor a.In general case, both SC **and** CC are described by equati**on**wherem cc = ma(η) (9)ds 2 cc = (dη) 2 − (dx k ) 2 (10)ρ(a) = ρ 0(a ′ ) 2 = ρ(a), (11)∑J=−2,0,1,4Ω J a J (12)is the c**on**formal energy density c**on**nected with the SC **on**e by the transformati**on**ρ(a) = a 4 ρ SC (a) , (13)4

factor a 4 = a 3 a arises due to differences **of** length **and** mass units in SC **and** CC, Ω J is partial energydensity marked by index J running a set **of** values J = −2 (rigid), J = 0 (radiati**on**), J = 1 (mass),∑**and** J = 4 (Λ-term) in corresp**on**dence with a type **of** matter field c**on**tributi**on**s; here Ω J = 1is assumed.In terms **of** the st**and**ard cosmological definiti**on**s **of** the redshiftthe density parameter Ω c (z) =∑J=−2,0,1,4Ω c (z) = Ω rig (1 + z) 2 + Ω rad +where Ω rig = Ω −2 , Ω rad = Ω 0 , Ω m = Ω 1 , Ω Λ = Ω 2 .Then the equati**on** (11) takes the formJ=−2,0,1,41 + z ≡ 1a(η) . (14)Ω J a J in Eq. (12) takes the formΩ m(1 + z) + Ω Λ(1 + z) 4 , (15)H 0dηdz = 1(1 + z) 2 1√Ωc (z) , (16)**and** determines the dependence **of** the c**on**formal time **on** the redshift factor. This equati**on** is valid als**of**or the c**on**formal time - redshift relati**on** in the SC where this c**on**formal time is used for descripti**on****of** a light ray.A light ray traces a null geodesic, i.e. a path for which the c**on**formal interval (ds L ) 2 = 0 thussatisfying the equati**on** dr/dη = 1. As a result we obtain for the coordinate distance as a functi**on** **of**the redshiftH 0 r(z) =∫ z0dz ′(1 + z ′ ) 2 1√Ωc (z ′ ) . (17)The equati**on** (17) coincides with the similar relati**on** between coordinate distance **and** redshift in SC.In the comparis**on** with the stati**on**ary space in SC **and** stati**on**ary masses in CC, a part **of** phot**on**sis lost. To restore the full luminosity in both SC **and** CC we should multiply the coordinate distanceby the factor (1 + z) 2 . This factor comes from the evoluti**on** **of** the angular size **of** the light c**on**e **of**emitted phot**on**s in SC, **and** from the increase **of** the angular size **of** the light c**on**e **of** absorbed phot**on**sin CC. However, an observable length in SC c**on**tains an additi**on**al factor (1 + z) −1in comparis**on** with measurable distances in CC∫ dtl SC = aa = r1 + z . (18)l CC =∫ dta = r1 + z . (19)(that coincide with the coordinate **on**es).Thus, we obtain the relati**on**s for observable lengths in SC **and** CC, respectivelyl SC (z) = (1 + z)r(z) , (20)l CC (z) = (1 + z) 2 r(z) , (21)simply because **of** l SC (z) = al CC (z). It means that the observati**on**al data are described by differentmanners in SC **and** CC, since functi**on**s H 0 l(z) have different representati**on**s in SC **and** CC modelstherefore, observati**on**al data can be fitted with different parameters **of** the functi**on**s as we willdem**on**strate below.5

CC optimalSC optimalCC rigidCC matterCC lambdaCC radFigure 1: µ(z)-dependence for cosmological models in SC **and** CC. The data points include 186 SN Ia(the ”gold” **and** ”silver” sample) used by the cosmological supernova HST team [30]. For a referencewe use the best fit for the flat st**and**ard **cosmology** model with Ω m = 0.27, Ω Λ = 0.73 (the thick dashedline), the best fit for CC is shown with the thick solid line. For this CC model we assume Ω m ≥ 0.3.1 Magnitude-Redshift Relati**on**Typically to test cosmological **theories** **on**e should check a relati**on** between an apparent magnitude**and** a redshift. In both SC **and** CC models it should be valid the effective magnitude-redshift relati**on**:µ(z) ≡ m(z) − M = 5 log [H 0 l(z)] + M, (22)where m(z) is an observed magnitude, M is the absolute magnitude, M is a c**on**stant with recentexperimental data for distant SNe. Values **of** µ i , z i **and** σ i could be taken from observati**on**s **of**a detected supernova with index i (σ 2 i is a dispersi**on** for the µ i evaluati**on**). Since we deal withobservati**on**al data we should choose model parameters to satisfy an array **of** relati**on**s (22) by thebest way because usually, a number **of** relati**on**s is much more than a number **of** model parameters**and** there are errors in both theory **and** observati**on**s (as usual we introduce indices for the relati**on**scorresp**on**ding to all objects). Typically, χ 2 -criterium is used to solve the problem, namely, we calculateχ 2 = ∑ i(µ theori − µ i ) 2, (23)σ 2 iwhere µ theori are calculated for given z i with the assumed theoretical model **and** after that we canevaluate the best fit model parameters minimizing χ 2 -functi**on**.However, in principle, other **alternative**s for **cosmology** are not completely ruled out, for example,a c**on**formal cosmological model was discussed in papers [23, 24]. Using ”gold” **and** ”silver” 186 SNeIa [30] we c**on**firm in general **and** clarify previous c**on**clusi**on**s about CC model parameters [31], d**on**eearlier with analysis **of** smaller sample **of** SNe Ia data [23, 24] that the pure flat rigid CC model couldfit the data relatively well since ∆χ 2 ≈ 44.3 (or less than 20 %) in respect **of** the st**and**ard **cosmology**flat model with Ω m = 0.28. Other pure flat CC models should be ruled out since their χ 2 values aretoo high. For the total sample, if we c**on**sider CC models with a ”realistic” c**on**straint 0.2 Ω m 0.3based **on** other astr**on**omical or cosmological arguments except SNe Ia data, we c**on**clude that thest**and**ard **cosmology** flat model with Ω m = 0.28 is still preferable in respect to the fits for the CCmodels but the preference is not very high (about 5 % in relative units **of** χ 2 value), so the CC modelscould be adopted as acceptable **on**es taking into account possible sources **of** errors in the sample **and**systematics.6

Figure 2: The parameter β as a functi**on** **of** n for fourth order **gravity**.that the st**and**ard general relativity plus DM **and** DE may be distinguished from R n approaches withgravitati**on**al microlensing [40], but Solar system data (planetary orbital periods, in particular) putsevere c**on**straints **on** parameters **of** the the **theories** [41].4.1 Planetary **on**straints**on**al c**on**sequences **of** fourth order **gravity** which, after fixing the coreradius r c ≃ 1 A.U., still depends **on** the parameter β in the weak field approximati**on**.A c**on**straint **on** the fourth order **gravity** theory can be obtained from the moti**on** **of** the solar systemplanets. Let us c**on**sider as a toy model a planet moving **on** circular orbit (**of** radius r) around the Sun.From Eq. (25), the planet accelerati**on** a = −∂Φ(r)/∂r is given by[a = − Gm ( ) β ( ) ] β r r2r 2 1 + − β . (27)r c r cAccordingly, the planetary circular velocity v can be evaluated **and**, in turn, the orbital period P isgiven byP = P K√2[1 +( rr c) β− β( rr c) β] −1/2, (28)where P K = [4π 2 r 3 /(Gm)] 1/2 is the usual Keplerian period. In order to compare the orbital periodpredicted by the fourth order theory with the Solar System observati**on**s, let us define the quantity∆P= |P − P K|= |f(β, r c ) − 1| (29)P K P K8

eing f(β, r c ) the factor appearing **on** the right h**and** side **of** equati**on** (28) **and** multiplying the usualKeplerian period.There is a questi**on** about a possibility to satisfy the planetary period c**on**diti**on** – vanishing theEq. (29) – with β parameter which is significantly different from zero. Vanishing the right h**and** side**of** Eq. (29) we obtain the relati**on**ln (1 − β)ln r = ln r c − , (30)βso that Eq. (30) should be satisfied for all the planetary radii. This is obviously impossible since thefourth order theory defines β as a parameter, while the specific system under c**on**siderati**on** (the Solarsystem in our case) allows us to specify the r c parameter. Hence the right h**and** side **of** Eq. (30) isfixed for the Solar system, implying that it is impossible to satisfy Eq. (30) even with two (or more)different planetary radii.Just for illustrati**on** we presented the functi**on** f(β, r c ) as a dependence **on** β parameter for fixed r c**and** planetary radii r (see Figs. in paper [41]). As **on**e can see from Eq. (30) (**and** Figs. in paper [41]as well) for each planetary radius r > r c there is β ∈ (0, 1) satisfying Eq. (30), but the β value depends**on** fixed r c **and** r, so that they are different for a fixed r c **and** different r. Moreover, if we have at least**on**e radius r ≤ r c , there is no soluti**on** **of** Eq. (30). Both cases imply that the β parameter should bearound zero.As **on**e can note, **on**ly for β approaching zero it is expected to recover the value **of** the Keplerianperiod. In the above menti**on**ed figure, the calculati**on** has been performed for the Earth orbit (i.e.r = 1 A.U.).Current observati**on**s allow also to evaluate the distances between the Sun **and** the planets **of** theSolar System with a great accuracy. In particular, differences in the heliocentric distances do notexceed 10 km for Jupiter **and** amount to 180, 410, 1200 **and** 14000 km for Saturn, Uranus, Neptune**and** Pluto, respectively [42]. Errors in the semi-major axes **of** the inner planets are even smaller(see e.g. Table 2 in the paper [43]) so that the relative error in the orbital period determinati**on** isextremely low. As an example, the orbital period **of** Earth is T = 365.256363051 days with an error**of** ∆T = 5.0 × 10 −10 days, corresp**on**ding to a relative error **of** ∆T/T less than 10 −12 . These valuescan be used in order to c**on**strain the possible values **of** both the parameters β **and** r c introduced bythe fourth order **gravity** theory. This can be d**on**e by requiring that ∆P/P K ∆T/T so that, in thecase **of** Earth, |f(β, r c ) − 1| 10 −12 which can be solved with respect to β **on**ce the r c parameter hasbeen fixed to some value. For r c = 1 AU **and** r c = 10 4 AU (i.e. the two limiting cases c**on**sidered inthe paper [40]) we find the allowed upper limits **on** the β parameter to be 4.0 × 10 −12 **and** 3.9 × 10 −13 ,respectively (since ∆P/P K = ∆β[−1 + ln (r/r c )]/4).A more precise analysis which takes into account the planetary semi-major axes **and** eccentricitiesleads to variati**on**s **of** at most a few percent, since the planetary orbits are nearly circular. Therefore,in spite **of** the fact that orbital periods **of** planets are not generally used to test **alternative** **theories****of** **gravity** (since it is taken for granted that the weak field approximati**on** **of** these **theories** gives theNewt**on**ian limit), we found that these data are important to c**on**strain parameters **of** the fourth order**gravity** theory.4.2 Discussi**on**GR **and** Newt**on**ian theory (as its weak field limit) were verified by a very precise way at different scales.There are observati**on**al data which c**on**strain parameters **of** **alternative** **theories** as well. As a result,the parameter β **of** fourth order **gravity** should be very close to zero (it means that the gravitati**on**altheory should be very close to GR). In particular, the β parameter values c**on**sidered for microlensing[40], for rotati**on** curves [34] **and** cosmological SN type Ia [35] are ruled out by solar system data.No doubt that **on**e could also derive further c**on**straints **on** the fourth order **gravity** theory byanalyzing other physical phenomena such as Shapiro time delay, frequency shift **of** radio phot**on**s [44],laser ranging for distant objects in the solar system, deviati**on**s **of** trajectories **of** celestial bodies fromellipses, parabolas **and** hyperbolas **and** so **on**. But our aim was **on**ly to show that **on**ly β ≃ 0 valuesare not in c**on**tradicti**on** with solar system data in spite **of** the fact that there are a lot **of** speculati**on**sto fit observati**on**al data with β values significantly different from zero.9

5 C**on**clusi**on**sIn c**on**clusi**on** we note that, unfortunately, even well-informed Russian (**and** other) authors did not citeRelikt-1 results in papers **on** **cosmology**, where the COBE anisotropy result was quoted as the **on**lyexperiment discovered the phenomen**on**. It means the Nobel prize winner (1978) P. L. Kapitza 6 wrotequite correctly about this kind **of** problems in 1946: ”... Our main nati**on**al defect is an underestimati**on****of** our powers **and** overestimati**on** **of** foreign **on**es. So, an extra modesty is much more defective thanan extra self-c**on**fidence... Very **of**ten a cause **of** unused innovati**on**s is that usually we underestimateour own discoveries **and** overestimate foreign **on**es...”For CC model fits calculated with SNe Ia data, in some sense, a rigid equati**on** **of** state couldsubstitute the Λ-term (or quintessence) in the Universe c**on**tent. The rigid matter can be formed by afree massless scalar field [31].Solar system data put severe limits **on** parameters **of** **alternative** **theories** **of** **gravity** [41], thus,the Solar system limits a **gravity** law not **on**ly at scales about several A.U., but also at cosmologicaldistances, in this class **of** **alternative** **theories** **of** **gravity**.Acknowledgement. AFZ acknowledges GAS@BS-2007 c**on**ference organizers, especially pr**of**.P. Fiziev, for the hospitality **and** a financial support.References[1] Gamow, G. (1946) Exp**and**ing Universe **and** Orgin **of** Elements, Phys. Rev. 70, 572–573.[2] Gamow, G. (1948) The Origin **of** Elements **and** the Separati**on** **of** Galaxies, Phys. Rev. 74, 505–506.[3] Gamow, G. (1948) Evoluti**on** **of** the Universe, Nature 162, 680.[4] Alpher, R. A. (1948) A Neutr**on** – Capture Theory **of** the Formati**on** **and** Relative Abundance **of** Elements,Phys. Rev. 74, 1577–1589.[5] Gamow, G. (1956) The physics **of** the exp**and**ing universe, Vistas in Astr**on**omy 2, 1726–1732.[6] Penzias, A. A., Wils**on**, R. W. (1965) A Measurement **of** Excess Antenna Temperature at 4080 Mc/s,Astrophys. J. 142, 419–420.[7] Dicke, R.H., Peebles, P. J. E., Roll, P. G., Wilkins**on**, D. T. (1965) Cosmic Black-Body Radiati**on**,Astrophys. J. 142, 414–419.[8] C**on**klin, E.-K. (1969) Velocity **of** the Earth with Respect to the Cosmic Background Radiati**on**, Nature,222, 971–972.[9] Henry, P.S. (1971) Isotropy **of** the 3 K Background, Nature, 231, 516–518.[10] Corey, B.E., Wilkins**on**, D. T. (1976) A Measurement **of** the Cosmic Mirowave Background Anisotropy at19 GHz Bull. Amer. Astr**on**. Soc., 8, 351.[11] Smoot, G.F., Gorenstein, M.V., Muller, R. A. (1977) Detecti**on** **of** Anisotropy in the Cosmic BlackbodyRadiati**on**, Phys. Rev. Lett., 39, 898–891.[12] Fabbri, R., Giudi, I., Melchiorri, F., Natale, V. (1980) Measurement **of** the Cosmic-Background Large-Scale Anisotropy in the Millimetric Regi**on**, Phys. Rev. Lett., 44, 1563–1566.[13] Boughn, S.P., Cheng, E. S., Wilkins**on**, D. T., (1981) Dipole **and** quadrupole anisotropy **of** the 2.7 Kradiati**on** Astrophys. J., 243, L113–L117.[14] Vernov, S. N. et al. (eds.) (1977) Problems **of** Solar activity **and** space system ”Prognoz” Nauka, Moscow,in Russian).[15] Klypin, A. A., Sazhin, M. V., Strukov, I. A., Skulachev, D.P. (1987), Limits **on** Microwave BackgroundAnisotropies - the Relikt Experiment, Sov. Astr**on**. Lett. 13, 104–107.6 He was awarded by the Nobel prize for his experimental discovery **of** superfluidity **and** received the prize in the sameyear as Penzias **and** Wils**on** got. He was the favorite Rutherford’s student **and** worked for a l**on**g time in UK, therefore,he knew Western **and** Russian mentalities very well.10

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