A framework for the next generation of stationary cosmological models Yves-Henri Sanejouand∗ Facult´edes Sciences et des Techniques, Nantes, France. August 11th, 2021 Abstract Such predictions have been backed by numer- ous observations. For instance, the expected time- According to a new tired-light cosmological model, dilation of remote events has been found in the light where H(z) = H0(1 + z), the number density of curves of supernovae Ia [5, 6, 7, 8], a thermal radi- ◦ galaxies has been nearly constant over the last ation at a temperature of T0 = 2:7 K has been ob- 10 Gyr, at least, meaning that, as far as galaxy served [9, 10] and its redshift dependence has been counts are concerned, the Universe has been sta- confirmed [11, 12, 13]. tionary. In this context, an analysis of the lumi- However, several clouds are still obscuring the nosity distances of quasars and supernovae Ia also brilliance of ΛCDM [14, 15, 16], the so-called "con- shows that the Universe is far from being as trans- cordance cosmology" [17]. Among them, the Hub- parent as assumed nowadays, the photon lifetime ble tension seems to attract most of the attention along the line-of-sight being one third of the Hub- [18]. Still, other clouds may prove darker [19, 20]. ble time. The tired-light model advocated in the In particular, ΛCDM is based on a "cosmic trinity" present study would be falsified if, for instance, the [21] of three essential ingredients with weird prop- time-dilation of remote events were shown to have erties and for which there is no direct evidence, a general character, that is, if it were observed for namely, an early stage of accelerated expansion phenomenons other than the light-curves of super- [22, 23], a dark matter and an energy components novae Ia. of unknown nature [16, 24], both accounting for ≈ 95% of the matter-energy content of the Universe Keywords: Alternative cosmologies, Cosmic opac- [25]. ity, Angular distance, Luminosity distance, Dis- In other words, according to ΛCDM, the domi- tance duality. nant forms of matter-energy are of a different na- ture on Earth and far away. Though such an hy- pothesis was taken for granted in the ancient times, Introduction it is the opposite hypothesis that has proven fruit- ful since the Renaissance, namely, that what is ob- The family of cosmologies initiated by Georges served on Earth is representative of what is found Lemaitre [1] proved able to make challenging pre- in the rest of the Universe. Given the numerous arXiv:2005.07931v4 [astro-ph.CO] 11 Aug 2021 dictions. Among them: luminosity distances are successes of the later hypothesis, it seems reason- larger than angular ones by a factor of (1 + z)2 [2]; able to push it forward once more. all remote events look slower than local ones by a factor of (1 + z) [3]; there is an isotropic radiation However, it is not obvious to build from scratch with the spectrum of a blackbody at a temperature a new cosmology able to compete with the result of the work of several generations of brilliant scien- of T0(1 + z), T0 being its local temperature [4]. tists. Also, given the huge time and distance scales ∗[email protected] involved in cosmological problems, key physical in- 1 gredients may still be missing, like a tiny variation of quantities nowadays assumed to be constant (e.g. [26, 27, 28]). So, as a preliminary step, it may prove useful to pinpoint a set of ingredients which could serve as a basis for the development of such a new cosmology. This is the main goal of the present study. Main hypothesis The tired-light model As proposed a while ago [29, 30], let us assume that photons can not fly away for ever. However, instead of interpreting electromagnetic radiation as an interaction between a source and an absorber, Figure 1: H(z) as a function of redshift, as ob- let us posit that photons have all the same maxi- tained with the cosmic chronometer method. Dot- mum range, dH , due to a loss of their energy such ted line: linear fit. that: hνobs = hν0 − fγ dT (1) already been proposed a number of times (e.g. [32, 33, 34, 35, 36, 37, 38, 39]), note that the hy- where ν0 is the frequency of the photon when it is pothesis that photons may all have same range has emitted, νobs, its frequency when it is observed, dT , the distance between its source and the observer, h been, to my knowledge, little considered so far. being the Planck constant. h νobs = 0 when dT = dH . So, fγ = ν0 and eqn Consistency with H(z) data dH 1 can also be written as follows: Let us now assume that, as postulated by Einstein dT [40], and as checked in various contexts (e.g. [41, νobs = ν0(1 − ) dH 42, 43, 44, 45, 46, 47]), the speed of light is constant and that delays, due for instance to the Shapiro that is: z d effect [48], can be neglected in such a way that: = T (2) 1 + z dH dT ≈ c0∆t (4) Thus, when z 1: where ∆t = t0−t is the photon time-of-flight, t0 and t being the observer and cosmic times, respectively. dT z ≈ Thus, with eqn 3, eqn 2 becomes: dH z H0 So, assuming that: = dT (5) 1 + z c0 c d = 0 (3) that is, with eqn 4 [49]: H H 0 z = H ∆t (6) yields: 1 + z 0 H0 z ≈ dT As a consequence: c0 @z which is the relationship anticipated by Lemaitre = −H (1 + z)2 (7) @t 0 [1] and further confirmed by Hubble [31], H0 being @z the Hubble constant, c0, the speed of light. Values of @t can be determined with the cosmic Though the idea that the Lemaitre-Hubble law chronometer method, where t is obtained by mea- is the result of some tired-light mechanism has suring the age of passively evolving galaxies [50, 51]. 2 They are usually provided through H(z), which is defined as follows [52]: 1 @z H(z) = − 1 + z @t Thus, with eqn 7: H(z) = H0(1 + z) (8) It is indeed well known that, as illustrated in Fig- ure 1, observational data [53] are consistent with a linear relationship [49, 54]. As a matter of fact, eqn 8 is also a prediction of linear coasting cosmologies [55, 56], like the Rh = ct cosmology developed by Fulvio Melia and his collaborators [57] who have claimed that, compared to ΛCDM, it is favored by Figure 2: Cumulative counts of galaxies as a func- various model selection criteria [58, 59]. tion of redshift. Top and bottom: sources of long gamma-ray bursts (GRBs) with a low (top) or high (bottom) redshift completeness level. Middle: Counts of galaxies galaxies in the Hubble Ultra Deep Field (HUDF), with robust spectroscopic redshifts. Dotted lines: n(dT ), the cumulative count of galaxies as a func- single-parameter fits (eqn 12), when the corre- tion of the light-travel distance, is such that: sponding number density of galaxies is assumed to be constant (N = 0), for z ≤ 2 (HUDF) or Z dT 2 n(dT ) = 4πN(r)r dr (9) 3 (GRBs). 0 where N(r) is the number density of galaxies at Least-square fitting, for z ≤ 3, of the cumulative distance r. count of 52 Swift long gamma-ray bursts (GRBs) Let us assume that N(∆t), the number density of from a carefully selected sub-sample with a redshift galaxies as a function of the photon time-of-flight, completeness level of 90% [60] yields N = 0.12 ± evolves slowly enough, so that: 0.09 (nst = 104 ± 6), confirming that the evolu- _ tion of the number density of GRBs has been slow, N(∆t) ≈ N0 + N∆t (10) −1 with respect to the Hubble time (H0 ), as assumed where N0 is the local number density, N_ being the above (eqn 10). time derivative of N(∆t). With eqn 4 and 10, eqn As a matter of fact, if the number density of _ 9 yields: GRBs does not vary as a function of redshift (N = 0, that is, N = 0), the root-mean-square of ! _ 4 3 3 N dT the residuals is 0.97 (nst = 113 ± 1), instead of n(dT ) = πdT N0 1 + (11) 3 4 N0 c0 0.96, meaning that both fits are equally consistent with observational data. When the 341 Swift long which becomes, with eqn 3 and 5: (t90 ≥ 0.8 s [61]) GRBs with a redshift known with fair accuracy are considered1 the root-mean-square z3 z of the residuals is higher, namely 5.6 (nst = 646 n(z) = nst 1 + N (12) (1 + z)3 1 + z ± 1), maybe as a consequence of the much lower redshift completeness level (redshifts are known for where: 4 only 30% of the GRBs detected by Swift [62]). n = πd3 N st 3 H 0 On the other hand, fitting the cumulative count and: 1As provided on the Neil Gehrels Swift Observatory _ 3 N −1 web page, (https://swift.gsfc.nasa.gov/archive/grb table), N = H0 (13) on May 2020, 12th. 4 N0 3 of the 169 galaxies in the Hubble Ultra Deep Field are roughly consistent with constant angular sizes (HUDF) with a robust spectroscopic redshift2 [63], above z ≈ 2:5, with a mean half-light radius of for z ≤ 2, yields a similar root-mean-square of the 0.2400 at z ≈ 4 [74].