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- 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- ◦ 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 served [9, 10] and its dependence has been counts are concerned, the 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 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 and an components novae Ia. of unknown [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 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 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 . 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]. Note that, according to eqn residuals, namely, of 5.5 (nst = 513 ± 3). 14, if H0 ≈ 70 km/s/Mpc [25, 75, 76, 77, 78], the So, as shown in Figure 2, when the number den- mean half-radius of Lyman-break galaxies at z ≈ 4 sity of galaxies is assumed to be constant (N˙ = 0), is s ≈ 4 kpc, that is, the order of magnitude of the eqn 12 allows for a fair single-parameter fit of the optical radius of a galaxy in the local Universe. observational data, up to z ≈ 2 at least. On the contrary, with ΛCDM as the background Since long GRBs occur in star-forming galaxies cosmology, the average linear size of galaxies, hrGi, [64, 65], the fact that the number density of GRB is instead assumed to experience a strong evolution, α sources does not vary significantly as a function of with hrGi ∝ (1 + z) , α ranging between -1.2 and redshift suggests that the number density of star- -0.8 [79], hrGi being for instance up to six times forming galaxies does not as well, as already indi- smaller at z = 3.2 than at z = 0 [80]. cated by previous studies [66, 67]. On the other hand, the fact that the number density of galax- ies is also found nearly constant in the HUDF fur- Luminosity distance ther suggests that the number density of quiescent galaxies follows the same trend. In a , dL, the luminosity distance, is So, according to the above analysis, as far as expected to have a form like: galaxy number densities are concerned, the Uni- verse seems to have been stationary, at least over 1 1 τ(z) dL = dT (1 + z) 2 e 2 (15) the last 10 Gyr (z ≤ 2). Note that, with ΛCDM as the background cos- 1 where the (1 + z) 2 term corresponds to the energy mology, the star-formation rate density is instead loss of the photons during their travel, while τ(z) assumed to have a complex history, namely, to have denotes the opacity between the source and the ob- peaked approximately 3.5 Gyr after the , server [81], herein assumed to be mostly due to a and to have declined exponentially at later times single physical phenomenon. As such, it may prove [68]. well described by a relationship as simple as:

d Angular distance τ(z) = T (16) c0τγ Stationarity on galactic scales could mean that, as suggested in a number of previous studies (e.g. where τγ is the photon lifetime along the line-of- [69, 70, 71, 72, 73]), the space-time metric of the sight. Thus, with eqn 5, eqn 15 becomes: Universe is static. c0 z 1 1 z 2 H0τγ 1+z If so, dA = dT , dA being the angular distance, dL = √ e (17) H0 1 + z and θs, the angular size of a remote object, is so that: s θs ≈ Distance modulus dT where s is the actual size of the object. That is, With eqn 17, µ, the distance modulus:3 with eqn 5 [49]: µ = 5 log10(dL) + 25 −1 1 θs ≈ H0c0 s(1 + ) (14) z is as follows: This relationship noteworthy means that, for z  −1 z 1 z 1, θ → H c s. Interestingly, measurements of µ = 5 log √ + α + µ0 (18) s 0 0 10 1 + z H τ 1 + z the average angular size of Lyman-break galaxies 0 γ

2 c0 As found in Table 4 of reference [63]. where µ0 = 5 log + 25, with α = 2.5 log e. 10 H0 10

4 most part determined by their interaction with ob- jects belonging to the halos of galaxies. Thus, since, as found with the above analysis of galaxy counts, N˙ ≈ 0, eqn 16 can also be written as follows:

τ(z) ≈ σGN0dT (19)

where σG is the average cross section of the halo of a galaxy. On the other hand, for dT = dH , eqn 11 yields: n(dH ) N0 = 4 3 3 πdH

where n(dH ) is the total number of galaxies in the visible Universe. Thus, according to eqn 3, 16 and 19: Figure 3: The distance modulus of an homoge- 1 4πd2 neous sample of quasars, as a function of their σ = H G 3H τ n(d ) redshift. Each point (filled circles) is an average 0 γ H over 25 quasars, errorbars indicating the standard and since, as found above, 3H0τγ ≈ 1: deviation. Dashed line: ΛCDM. Plain line: two- 2 parameter least-square fit (eqn 18). 4πdH σG ≈ (20) n(dH ) Quasars Because there are ≈ 10,000 galaxies in the Hub- ble Extreme Deep Field (HXDF) [85], assuming In order to estimate τγ , a homogeneous sample of that most of the galaxies in this small area have 1598 quasars [82] with a distance modulus deter- been captured, and also that the HXDF is a rep- mined using their rest-frame X-ray and UV fluxes resentative enough sample of the sky, a rough esti- 11 [83, 84] was considered. To have fair estimates of mate can be proposed for n(dH ), namely, 4 10 . the standard deviation of these distance moduli, Then, according to eqn 3 and 20, if H0 ≈ 70 the dataset was sorted by increasing redshift val- 2 41 2 km/s/Mpc, σG = 4πrG ≈ 5 10 m , with a corre- ues and split into 64 groups of 25 quasars with sponding radius rG ≈ 7 kpc, that is, the order of similar redshift,4 the average redshift and distance magnitude of the optical radius of a galaxy in the modulus of each group being used for the present local Universe. analysis. Note that, being a sum over all the objects in- A least-square fit of these 64 average distance volved in the process responsible for the loss of pho- moduli yields H0τγ = 0.32 ±0.03 (µ0 = 43.1 ± 0.2). tons along the line-of-sight, the above value of σG Interestingly, as shown in Figure 3, eqn 18 matches is expected to be an effective one. 2 the observational data (reduced χdof = 0.05) over the whole redshift range, namely, up to z ≈ 5. Note that with ΛCDM as the background cos- Other checks mology (Ωm = 0.315 [25]), the quality of the fit is 2 Supernovae Ia similar (reduced χdof = 0.06). Though supernovae of type Ia (SN Ia) can not be The photon lifetime studied over a range of redshifts as wide as quasars, their luminosity distance can be determined with Let us now consider the hypothesis that the life- a much higher accuracy [86, 87]. Like above, the time of photons along the line-of-sight is for the 1048 SN Ia of the Pantheon sample were gathered in 42 groups of 25 SN Ia with similar redshifts,5 3 Where dL is in Mpc. 4With 23 quasars in the highest-redshift group. 5With 23 SN Ia in the highest-redshift group.

5 Figure 4: The distance modulus of the supernovae Ia of the Pantheon sample, as a function of their Figure 5: Compatibility with the cosmic dis- redshift. Each point (filled circles) is an average tance duality relation, as a function of redshift. over 25 SN Ia, errorbars indicating the standard Boxes: measurements obtained by studying early- deviation. Dashed line: ΛCDM. Plain line: as ex- type galaxies in the R (grey) and I (hatched) bands. pected with H0τγ = 0.32 (eqn 18). Filled circle: in the K band. Horizontal dashed line: minimum value expected within the frame of met- ric theories of gravity like ΛCDM, if there is no sig- the average redshift and distance modulus of each nificant luminosity evolution of early-type galaxies. group being used for the present analysis. Plain line: as expected with H0τγ = 0.32. As shown in Figure 4, with, as found above, H0τγ = 0.32, the absolute magnitude of a SN Ia being set 6 to M = −19.4 [88] (µ0 = 43.1), eqn 18 matches the 96]. For instance, by studying 34 early-type galax- distance moduli of the SN Ia pretty well (reduced ies from three clusters, with redshifts between 0.72 2 χdof = 0.04). Indeed, a least-square fit of these and 0.92, Lubin and Sandage found η(z) = 0.75– 7 8 data yields H0τγ = 0.317 ± 0.004 (µ0 + M = 23.74 0.89, in the I band, and η(z) = 0.57–0.71, in the ± 0.01). R one [92], in line with a previous value of η(z) = Note that with ΛCDM as the background cos- 0.82 ± 0.05,9 obtained by comparing the surface mology, the quality of the fit is similar (reduced brightness of the Abell 851 and Coma clusters [91]. 2 χdof = 0.04). Note that, with ΛCDM as the background cos- mology, such low values of η(z) are explained by as- Distance duality suming a strong luminosity evolution of early-type galaxies [92, 97, 98]. Let us write the cosmic distance duality relation as On the other hand, with eqn 5 and 17, eqn 21 follows [89]: yields: 3 1 1 z 2 − 2 2 H0τγ 1+z dL = η(z)dA(1 + z) (21) η(z) = (1 + z) e (22) In the case of this relationship, metric theories of As shown in Figure 5, according to eqn 22, with gravity, like ΛCDM, make the challenging predic- H0τγ = 0.32, for z ≈ 1, η(z) values are indeed well tion that η(z) ≥ 1 [90], with η(z) = 1 if there is no below one, as observed by Lubin and Sandage. loss of photon along the path between the source 7Corresponding to an exponent on (1 + z) between 3.06 and the observer [2]. and 3.55, instead of 4 [92]. Interestingly, measurements of η(z) tend to pro- 8Corresponding to an exponent on (1 + z) between 2.28 vide values that are below one [91, 92, 93, 94, 95, and 2.81, instead of 4 [92]. 9Corresponding to a surface brightness difference of 1.02 6 2 µ = mB − M, where mB is the apparent magnitude of ± 0.14 mag.arsec at z=0.4, when the k-correction is taken the . into account [91].

6 Towards a new cosmology Note that, in the context of the present study, the CMB temperature fluctuations could correspond to Old high redshift objects fluctuations of the photon time-of-flight (eqn 4), as a consequence of delays for instance due to the Remote objects are observed like they were at time Shapiro effect. t = t0 − ∆t. On the other hand, the oldest objects in our neighborhood, like HD140283, an extremely metal-deficient subgiant star nowadays known as Discussion the methuselah of stars, or the globular cluster NGC 6101, are ≈ 13–14 Gyr old [99, 100, 101] while, Are tired-light models still relevant ? for instance, the oldest object known at z ≈ 4, namely, APM 08279+5255, an exceptionally lumi- It has been claimed that theories where the nous, gravitationaly lensed, quasar, seems to be 2–3 Lemaitre-Hubble law is explained by a loss of en- Gyr old [102, 103, 104]. ergy of the photons during their travel are excluded Such observations suggest that, as predicted by [92], noteworthy because, as predicted by Lemaitre Lemaitre cosmologies, the oldest objects known cosmologies [3], the SN Ia light curves seem dilated started to emit light approximately at the same by a (1 + z) factor. time, at least Tf ≈ 14 Gyr ago. However, no such was found in the In the context of the present study, as far as the light curves of quasars [110, 111] or in duration oldest objects are concerned, To(z), their observed measures of GRBs [112, 113], casting doubts on age at a given redshift, is expected to be: a key prediction of Lemaitre cosmologies, namely, the generality of the phenomenon. To(z) = Tf − ∆t In the context of the present study, the time dila- that is, with eqn 6 [49]: tion of SN Ia light curves is instead expected to be 1 z either the signature of some evolutionary process To(z) = Tf − (23) H0 1 + z [114, 115, 116], or due to cosmology-dependent as- sumptions made during the analyses of the SN Ia Interestingly, according to eqn 23, if the age of light curves [58, 117]. HD140283 is assumed to provide a fair estimate for Tf , with H0 ≈ 70 km/s/Mpc, the oldest objects at z ≈ 4 are expected to have an age of To(4) ≈ 2.8 How can the Universe be stationary ? Gyr, in good agreement with the measured age of APM 08279+5255. The hypothesis that the Universe is stationary has Note that it has been underlined that ΛCDM been taken for granted for long (e.g. [69]), being can hardly cope with the estimated age of APM noteworthy put forward within the frame of steady- 08279+5255 [102, 105, 106, 107]. state cosmologies [118, 119, 120, 121]. However, in the later case, it was in a different context. In par- The cosmic microwave background ticular, the space-time metric was not considered as being static. The very existence of the Sunyaev-Zel’dovich effect On the other hand, herein, stationarity is ob- in the case of remote galaxy clusters [93, 108, 109] served on galactic scales, namely, when number strongly supports one of the major outcomes of densities of galaxies are considered. Thus, it has to Lemaitre cosmologies, namely, that the cosmic mi- be the result of some force able to cancel the effect crowave background (CMB) is a redshifted thermal of gravitational attraction, when the distance be- radiation, whose origin is far away (at z > 1). tween galaxies is of the order of magnitude of the Taken together with the fact that, as recalled average distance between neighboring ones. Note above, the oldest objects of the Universe seem to that, in order to counteract such an attraction, this have started to emit light ≈ 14 Gyr ago, it is tempt- force has to yield accelerations of the order of 10−12 ing to conclude that the thermal history of the Uni- m s−2, well below the lowest values measured on verse provided by Lemaitre cosmologies is the right Earth [122] or in the solar system [123, 124]. Note one. also that stationarity can be observed only if this

7 force has a distance dependence steeper than grav- Conclusion itation, a criterion that is not met by the force as- sociated to the cosmological constant [125]. The present study shows that, by combining a tired-light model where H(z) = H0(1 + z), in fair How are photons lost ? agreement with observational data (Fig. 1), with the hypothesis that the Universe is far from being H0τγ = 0.32 means that after ≈ 3 Gyr of travel as transparent as assumed nowadays [81, 134], it is half of the photons of a quasar or of a SN Ia are possible to obtain a two-parameter luminosity dis- missing. tance (eqn 17) able to match observations up to Absorption by massive amounts of dust in the z ≈ 5 (Fig. 3). halos of galaxies could be responsible for their loss Interestingly, for z  0, the corresponding cos- along the line-of-sight. However, since the quasar mic distance duality relation (eqn 21 and 22) differs luminosity-distances analyzed herein were deter- significantly from the prediction of metric theories mined by comparing their X-ray and UV fluxes of gravity like ΛCDM. Moreover, for z < 1 at least, [82], while SN Ia distance moduli were determined it seems in good agreement with observations (Fig. in the optical, such dust would have to be ”grey” 5). [126, 127] over a three-decade frequency range, at In this context, as far as galaxy number densities least. In this respect, MACHOs seem to be the are concerned, the Universe looks stationary, up to most likely candidates [128]. z ≈ 2 at least (Fig. 2). However, since the oldest On the other hand, it has been suggested that objects known started to emit light ≈ 14 Gyr ago, photons could have a finite lifetime [108, 129], e.g., possibly as a result of the cooling of a hot medium by decaying into lighter particles such as massive in a state of equilibrium, the thermal history of the neutrinos [130, 131], thus reducing their flux along Universe advocated within the frame of Lemaitre the line-of-sight. This would noteworthy mean that cosmologies may prove to be the right one. photons have a small, yet nonzero rest mass mγ of the order of: Acknowledgements h −68 mγ ≈ 2 ≈ 10 kgs τγ c0 I thank Guido Risaliti, for providing the dataset of quasar luminosity-distances, Gabriel Chardin, that is, well below current upper limits [132, 133]. Martin Lopez-Corredoira, Louis Marmet, Georges Paturel and Marcel Urban, for insightful discus- Can H0 be measured on Earth ? sions and comments. Eqn 6 suggests that, like in the case of most tired- light models, the Hubble constant could be mea- sured in laboratory experiments, as a frequency References drift proportional to the photon time-of-flight. [1] Lemaitre, G. (1927). Un Univers homog`ene Note however that such a measurement would be de masse constante et de rayon croissant a challenging one, the expected drift being of the rendant compte de la vitesse radiale des order of 10−18 s−1. n´ebuleusesextra-galactiques. Ann. Soc. Sci. But note also that this frequency drift may not Bruxelles 47, 49–59. occur on distance scales that small. As a matter of fact, in the course of the present study, the photon [2] Tolman, R.C. (1930). On the estimation of lifetime along the line-of-sight has been found to be distances in a curved universe with a non- 1 nearly 3 of the Hubble time. Such a numerical coin- static line element. Proc. Natl. Acad. Sci. cidence may prove significant. It could for instance USA 16(7), 511. mean that there is a causal relationship between the way photons interact with galaxies and their [3] Wilson, O.C. (1939). Possible applications of cosmological frequency drift, as claimed long ago supernovae to the study of the nebular red by [32]. shifts. Ap. J. 90, 634.

8 [4] Alpher, R.A. & Herman, R.C. (1949). Re- [13] Baranov, I., Jesus, J.F. & Lima, J.A. (2016). marks on the evolution of the expanding uni- Testing CCDM Cosmology with the Radia- verse. Phys. Rev. 75(7), 1089. tion Temperature-Redshift Relation. arXiv 1605, 04857. https://arxiv.org/abs/ [5] Kim, M., Lee, J., Matheson, T., McMahon, 1605.04857. R., Newberg, H., Pain, R. et al. (1996). Cos- mological time dilation using type Ia super- [14] Ostriker, J. & Steinhart, P.J. (1995). The novae as clocks. Nucl. Phys. B 51, 123–127. observational case for a low-density universe with a non-zero cosmological constant. Na- [6] Leibundgut, B., Schommer, R., Phillips, M., ture 377, 600–602. Riess, A., Schmidt, B., Spyromilio, J., Walsh, J., Suntzeff, N., Hamuy, M., Maza, J. et al. [15] Krauss, L.M. & Turner, M.S. (1995). The (1996). Time dilation in the light curve of the cosmological constant is back. Gen. Rel. distant type Ia supernova SN 1995K. Ap. J. Grav. 27(11), 1137–1144. https://arxiv. 466(1), L21–L24. org/abs/astro-ph/9504003. [7] Foley, R.J., Filippenko, A.V., Leonard, D.C., [16] Peebles, P.J.E. & Ratra, B. (2003). The cos- Riess, A.G., Nugent, P. & Perlmutter, S. mological constant and . Rev. (2005). A definitive measurement of time mod. phys. 75(2), 559. https://arxiv.org/ dilation in the spectral evolution of the abs/astro-ph/0207347. moderate-redshift type Ia supernova 1997ex. Ap. J. letters 626(1), L11. [17] Tegmark, M., Zaldarriaga, M. & Hamilton, A.J. (2001). Towards a refined cosmic concor- [8] Blondin, S., Davis, T.M., Krisciunas, K., dance model: Joint 11-parameter constraints Schmidt, B., Sollerman, J., Wood-Vasey, from the cosmic microwave background and W., Becker, A., Challis, P., Clocchiatti, A., large-scale structure. Phys. Rev. D 63(4), Damke, G. et al. (2008). Time dilation in 043007. type Ia supernova spectra at high redshift. Ap. J. 682(2), 724. [18] Di Valentino, E., Mena, O., Pan, S., Visinelli, L., Yang, W., Melchiorri, A., Mota, D.F., [9] Penzias, A.A. & Wilson, R.W. (1965). A Riess, A.G. & Silk, J. (2021). In the Realm measurement of excess antenna temperature of the Hubble tension − a Review of Solu- at 4080 mc/s. Ap. J. 142, 419–421. tions. arXiv 2103, 01183. https://arxiv. [10] Dicke, R.H., Peebles, P.J.E., Roll, P.G. & org/abs/2103.01183. Wilkinson, D.T. (1965). Cosmic black-body [19] L´opez-Corredoira, M. (2017). Tests and prob- radiation. Ap. J. 142, 414–419. lems of the standard model in cosmology. [11] Saro, A., Liu, J., Mohr, J., Aird, K., Ashby, Found. of Phys. 47(6), 711–768. https: M., Bayliss, M., Benson, B., Bleem, L., Boc- //arxiv.org/abs/1701.08720. quet, S., Brodwin, M. et al. (2014). Con- [20] Perivolaropoulos, L. & Skara, F. (2021). straints on the CMB temperature evolu- Challenges for ΛCDM: An update. arXiv tion using multiband measurements of the 2105, 05208. https://arxiv.org/abs/ Sunyaev–Zel’dovich effect with the South 2105.05208. Pole Telescope. Mon. Not. R. Astron. Soc. 440(3), 2610–2615. [21] Di Valentino, E., Melchiorri, A. & Silk, J. (2020). Planck evidence for a closed universe [12] Luzzi, G., G´enova-Santos, R., Martins, C., and a possible crisis for cosmology. Nature De Petris, M. & Lamagna, L. (2015). Con- Astronomy 4(2), 196–203. https://arxiv. straining the evolution of the CMB temper- org/abs/2003.04935. ature with SZ measurements from Planck data. J. Cosmol. Astrop. Phys. 2015(09), [22] Starobinsky, A.A. (1982). Dynamics of phase 011. https://arxiv.org/abs/1502.07858. transition in the new inflationary universe

9 scenario and generation of perturbations. [33] Nernst, W. (1937). Weitere pr¨ufungder an- Phys. Lett. B 117(3-4), 175–178. nahme eines station¨arenzustandes im weltall. Zeitschrift f¨urPhysik 106(9-10), 633–661. [23] Linde, A.D. (1982). A new inflationary uni- verse scenario: a possible solution of the hori- [34] Finlay-Freundlich, E. (1954). Red-shifts in zon, flatness, homogeneity, isotropy and pri- the spectra of celestial bodies. Proc. Phys. mordial monopole problems. Phys. Lett. B Soc. A 67(2), 192. 108(6), 389–393. [35] North, J.D. (1965). The measure of the uni- [24] Bartelmann, M. (2010). The dark universe. verse. A History of modern cosmology. Ox- Rev. Mod. Phys. 82(1), 331–382. ford University Press. [36] de Broglie, L. (1966). Sur le d´eplacement des [25] Aghanim, N., Akrami, Y., Ashdown, M., Au- raies ´emisespar un objet astronomique loin- mont, J., Baccigalupi, C., Ballardini, M., tain. Comptes Rendus Acad. Sci. Paris 263, Banday, A., Barreiro, R., Bartolo, N., Basak, 589–592. S. et al. (2020). Planck 2018 results. VI. Cos- mological parameters. Astronomy & Astro- [37] Pecker, J.C. & Vigier, J.P. (1988). A possible physics 641, A6. https://arxiv.org/abs/ tired-light mechanism. Apeiron 2, 13–15. 1807.06209. [38] Heymann, Y. (2014). The dichotomous cos- [26] Dirac, P.A. (1937). The cosmological con- mology with a static material world and stants. Nature 139(3512), 323–323. expanding luminous world. Progr. Phys. 10(3), 178–181. http://vixra.org/abs/ [27] Webb, J.K., Flambaum, V.V., Churchill, 1403.0927. C.W., Drinkwater, M.J. & Barrow, J.D. (1999). Search for time variation of the fine [39] Marmet, L. (2018). On the interpretation of structure constant. Phys. Rev. Lett. 82(5), spectral red-shift in astrophysics: A survey of 884–887. red-shift mechanisms-II. arXiv 1801, 07582. https://arxiv.org/abs/1801.07582. [28] Sanejouand, Y.H. (2009). About some possi- ble empirical evidences in favor of a cosmolog- [40] Einstein, A. (1905). Zur elektrodynamik be- ical time variation of the speed of light. Eu- wegter k¨orper. Annalen der physik 4, 891. rophys. Lett. 88(5), 59002. https://arxiv. [41] Krisher, T.P., Maleki, L., Lutes, G.F., Pri- org/abs/0908.0249. mas, L.E., Logan, R.T., Anderson, J.D. & Will, C.M. (1990). Test of the isotropy of the [29] Tetrode, H. (1922). Uber¨ den one-way speed of light using hydrogen-maser Wirkungszusammenhang der Welt. Eine frequency standards. Phys. Rev. D 42(2), Erweiterung der klassischen Dynamik. 731. Zeitschrift f¨urPhysik 10(1), 317–328. [42] Schaefer, B.E. (1999). Severe limits on vari- [30] Wheeler, J.A. & Feynman, R.P. (1945). In- ations of the speed of light with frequency. teraction with the absorber as the mechanism Phys. Rev. Lett. 82(25), 4964. https:// of radiation. Rev. Mod. Phys. 17(2-3), 157. arxiv.org/abs/astro-ph/9810479. [31] Hubble, E. (1929). A relation between [43] Antonini, P., Okhapkin, M., G¨okl¨u, E. & distance and radial velocity among extra- Schiller, S. (2005). Test of constancy of galactic nebulae. Proc. Natl. Acad. Sc. USA speed of light with rotating cryogenic opti- 15(3), 168–173. cal resonators. Phys. Rev. A 71(5), 050101. https://arxiv.org/abs/gr-qc/0504109. [32] Zwicky, F. (1929). On the redshift of spectral lines through interstellar space. Proc. Nat. [44] Tu, L.C., Luo, J. & Gillies, G.T. (2004). The Acad. Sc. USA 15(10), 773–779. mass of the photon. Rep. Prog. Phys. 68, 77.

10 [45] Eisele, C., Nevsky, A.Y. & Schiller, S. (2009). [54] Kumar, S. (2012). Observational constraints Laboratory test of the isotropy of light prop- on hubble constant and deceleration param- agation at the 10−17 level. Phys. Rev. Lett. eter in power-law cosmology. Mon. Not. R. 103(9), 090401. Astron. Soc. 422(3), 2532–2538.

[46] Nemiroff, R.J., Connolly, R., Holmes, J. & [55] Kolb, E.W. (1989). A coasting cosmology. Kostinski, A.B. (2012). Bounds on spectral Ap. J. 344, 543–550. from fermi-detected gamma ray bursts. Phys. Rev. Lett. 108(23), 231103. [56] Benoit-L´evy, A. & Chardin, G. (2012). Intro- https://arxiv.org/abs/1109.5191. ducing the Dirac-Milne universe. A&A 537, A78. https://arxiv.org/abs/1110.3054. [47] Williams, J.G., Turyshev, S.G. & Boggs, D.H. (2014). The past and present earth- [57] Melia, F. & Shevchuk, A.S.H. (2012). The moon system: the speed of light stays steady Rh = ct universe. Month. Not. Roy. Astron. as tides evolve. Planetary 3(1), 1–9. Soc. 419(3), 2579–2586. https://arxiv. org/abs/1109.5189. [48] Reasenberg, R., Shapiro, I., MacNeil, P., Goldstein, R., Breidenthal, J., Brenkle, J., [58] Melia, F. & Maier, R.S. (2013). Cosmic Cain, D., Kaufman, T., Komarek, T. & chronometers in the Rh = ct Universe. Zygielbaum, A. (1979). Viking relativity Month. Not. Roy. Astron. Soc. 432(4), 2669– experiment-verification of signal retardation 2675. https://arxiv.org/abs/1304.1802. by solar gravity. Ap. J. 234, L219–L221. [59] Melia, F. & Yennapureddy, M.K. (2018). [49] Sanejouand, Y.H. (2014). A simple Hubble- Model selection using cosmic chronometers like law in lieu of dark energy. arXiv 1401, with gaussian processes. J. Cosmol. Astrop. 2919. https://arxiv.org/abs/1401.2919. Phys. 2018(02), 034. https://arxiv.org/ abs/1802.02255. [50] Stern, D., Jimenez, R., Verde, L., Kamionkowski, M. & Stanford, S.A. [60] Salvaterra, R., Campana, S., Vergani, S.D., (2010). Cosmic chronometers: constraining Covino, S., D’Avanzo, P., Fugazza, D., the equation of state of dark energy. I: H(z) Ghirlanda, G., Ghisellini, G., Melandri, A., measurements. J. Cosmol. Astrop. Phys. Nava, L. et al. (2012). A complete sam- 2010(02), 008. ple of bright Swift long gamma-ray bursts. I. Sample presentation, luminosity function [51] Moresco, M., Verde, L., Pozzetti, L., and evolution. Ap. J. 749(1), 68. https: Jimenez, R. & Cimatti, A. (2012). New con- //arxiv.org/abs/1112.1700. straints on cosmological parameters and neu- trino properties using the expansion rate of [61] Bromberg, O., Nakar, E., Piran, T. & Sari, the universe to z ≈ 1.75. J. Cosmol. Astrop. R. (2013). Short versus long and collapsars Phys. 2012(07), 053. versus non-collapsars: A quantitative classifi- cation of gamma-ray bursts. Ap. J. 764, 179. [52] Jimenez, R. & Loeb, A. (2002). Constrain- https://arxiv.org/abs/1210.0068. ing cosmological parameters based on relative galaxy ages. Ap. J. 573(1), 37–42. [62] Gehrels, N., Chincarini, G., Giommi, P., Mason, K.O., Nousek, J.A. et al. (2004). [53] Yu, H., Ratra, B. & Wang, F.Y. (2018). Hub- The Swift gamma-ray burst mission. Ap. ble parameter and Baryon Acoustic Oscilla- J. 611(2), 1005. https://arxiv.org/abs/ tion measurement constraints on the Hubble astro-ph/0405233. constant, the deviation from the spatially flat ΛCDM model, the deceleration–acceleration [63] Rafelski, M., Teplitz, H.I., Gardner, J.P., transition redshift, and spatial curvature. Ap. Coe, D., Bond, N.A., Koekemoer, A.M., J. 856(1), 3. Grogin, N., Kurczynski, P., McGrath, E.J.,

11 Bourque, M. et al. (2015). UVUDF: Ultravio- [71] Crawford, D.F. (1993). A static stable uni- let Through Near-infrared Catalog and Pho- verse. Ap. J. 410, 488–492. tometric Redshifts of Galaxies in the Hub- [72] Lerner, E.J., Falomo, R. & Scarpa, R. (2014). ble Ultra Deep Field. A. J. 150(1), 31. UV surface brightness of galaxies from the https://arxiv.org/abs/1505.01160. local Universe to z ≈ 5. Int. J. Mod. Phys. D [64] Michalowski, M.J., Kamble, A., Hjorth, J., 23, 1450058. Malesani, D., Reinfrank, R., Bonavera, L., [73] Lerner, E.J. (2018). Observations contradict Cer´on,J.C., Ibar, E., Dunlop, J., Fynbo, J. galaxy size and surface brightness predictions et al. (2012). The optically unbiased GRB that are based on the expanding universe hy- host (TOUGH) survey. VI. Radio observa- pothesis. Mon. Not. R. Astron. Soc. 477(3), tions at z < 1 and consistency with typi- 3185–3196. cal star-forming galaxies. Ap. J. 755(2), 85. https://arxiv.org/abs/1205.4239. [74] Ferguson, H.C., Dickinson, M., Giavalisco, M., Kretchmer, C., Ravindranath, S., Idzi, [65] Japelj, J., Vergani, S., Salvaterra, R., R., Taylor, E., Conselice, C.J., Fall, S.M., D’Avanzo, P., Mannucci, F., Fernandez- Gardner, J.P. et al. (2004). The size evo- Soto, A., Boissier, S., Hunt, L., Atek, H., lution of high-redshift galaxies. Ap. J. Let- Rodr´ıguez-Mu˜noz,L. et al. (2016). Are long ters 600(2), L107. https://arxiv.org/ gamma-ray bursts biased tracers of star for- abs/astro-ph/0309058. mation? Clues from the host galaxies of the Swift/BAT6 complete sample of bright [75] Riess, A.G., Macri, L.M., Hoffmann, S.L., LGRBs-II. Star formation rates and metal- Scolnic, D., Casertano, S., Filippenko, A.V., licities at z < 1. A&A 590, A129. https: Tucker, B.E., Reid, M.J., Jones, D.O., Silver- //arxiv.org/abs/1604.01034. man, J.M. et al. (2016). A 2.4% determina- tion of the local value of the Hubble constant. [66] Borch, A., Meisenheimer, K., Bell, E.F., Rix, Ap. J. 826(1), 56. https://arxiv.org/abs/ H.W., Wolf, C., Dye, S., Kleinheinrich, M., 1604.01424. Kovacs, Z. & Wisotzki, L. (2006). The stellar masses of 25000 galaxies at 0.2 < z < 1.0 [76] Paturel, G., Teerikorpi, P. & Baryshev, Y. estimated by the COMBO-17 survey. A&A (2017). Hubble law: measure and inter- 453(3), 869–881. pretation. Found. Phys. 47(9), 1208–1228. https://arxiv.org/abs/1801.00128. [67] Brammer, G.B., Whitaker, K.E., van [77] Riess, A.G., Casertano, S., Yuan, W., Macri, Dokkum, P.G., Marchesini, D., Franx, M., L., Anderson, J., MacKenty, J.W., Bowers, Kriek, M., Labbe, I., Lee, K.S., Muzzin, A., J.B., Clubb, K.I., Filippenko, A.V., Jones, Quadri, R.F. et al. (2011). The number den- D.O. et al. (2018). New parallaxes of galac- sity and mass density of star-forming and qui- tic cepheids from spatially scanning the hub- escent galaxies at 0.4 < z < 2.2. Ap. J. 739, ble space telescope: Implications for the hub- 24. https://arxiv.org/abs/1104.2595. ble constant. Ap. J. 855(2), 136. https: [68] Madau, P. & Dickinson, M. (2014). Cos- //arxiv.org/abs/1801.01120. mic star-formation history. Annual Review [78] Wong, K.C., Suyu, S.H., Chen, G.C., Rusu, of Astron. Astrophys. 52, 415–486. https: C.E., Millon, M., Sluse, D., Bonvin, V., . //arxiv.org/abs/1403.0007 Fassnacht, C.D., Taubenberger, S., Auger, [69] Einstein, A. (1917). Kosmologische betra- M.W. et al. (2020). H0LiCOW XIII. A chtungen zur allgemeinen Relativitatstheorie. 2.4% measurement of H0 from lensed quasars: Sitz. Preuss. Akad. Wiss. 1, l42–l52. 5.3σ tension between early and late-Universe probes. Mon. Not. R. Astron. Soc. 498, [70] LaViolette, P.A. (1986). Is the universe really 1420–1439. https://arxiv.org/abs/1907. expanding? Ap. J. 301, 544–553. 04869.

12 [79] Allen, R.J., Kacprzak, G.G., Glazebrook, 517(2), 565–586. https://arxiv.org/abs/ K., Labb´e,I., Tran, K.V.H., Spitler, L.R., astro-ph/9812133. Cowley, M., Nanayakkara, T., Papovich, C., Quadri, R. et al. (2017). The Size Evolu- [88] Suzuki, N., Rubin, D., Lidman, C., Aldering, tion of Star-forming Galaxies since z ≈ 7 Us- G., Amanullah, R. et al. (2012). The Hubble ing ZFOURGE. Ap. J. Letters 834(2), L11. Space Telescope Cluster Supernova Survey. https://arxiv.org/abs/1612.05262. V. Improving the Dark-energy Constraints above z > 1 and Building an Early-type- [80] L´opez-Corredoira, M. (2010). Angular size hosted Supernova Sample. Ap. J. 746(1), 85. test on the expansion of the universe. Int. J. https://arxiv.org/abs/1105.3470. Mod. Phys. D 19(03), 245–291. [89] Holanda, R.F.L., Lima, J.A.S. & Ribeiro, [81] Holanda, R. & Busti, V. (2014). Probing cos- M.B. (2010). Testing the Distance–Duality mic opacity at high redshifts with gamma- Relation with Galaxy Clusters and Type Ia ray bursts. Phys. Rev. D 89(10), 103517. Supernovae. Ap. J. letters 722(2), L233. https://arxiv.org/abs/1402.2161. https://arxiv.org/abs/1005.4458.

[82] Risaliti, G. & Lusso, E. (2019). Cosmolog- [90] Bassett, B.A. & Kunz, M. (2004). Cos- ical constraints from the Hubble diagram of mic distance-duality as a probe of exotic quasars at high redshifts. Nat. Astr. 3(3), physics and acceleration. Phys. Rev. D 272. https://arxiv.org/abs/1811.02590. 69(10), 101305. https://arxiv.org/abs/ astro-ph/0312443. [83] Risaliti, G. & Lusso, E. (2015). A hub- ble diagram for quasars. Ap. J. 815(1), 33. [91] Pahre, M.A., Djorgovski, S.G. & De Car- https://arxiv.org/abs/1505.07118. valho, R.R. (1996). A Tolman surface bright- ness test for universal expansion and the evo- [84] Lusso, E. & Risaliti, G. (2016). The tight lution of elliptical galaxies in distant clusters. relation between X-ray and ultraviolet lu- The Astrophysical Journal Letters 456(2), minosity of quasars. Ap. J. 819(2), 154. L79. https://arxiv.org/abs/astro-ph/ https://arxiv.org/abs/1602.01090. 9511061.

[85] Illingworth, G., Magee, D., Oesch, P., [92] Lubin, L.M. & Sandage, A. (2001). The Tol- Bouwens, R.J., Labb´e, I., Stiavelli, M., man surface brightness test for the reality of Van Dokkum, P., Franx, M., Trenti, M., Car- the expansion. IV. A measurement of the Tol- ollo, C.M. et al. (2013). The HST eXtreme man signal and the luminosity evolution of deep field (XDF): combining all ACS and early-type galaxies. A. J. 122(3), 1084. WFC3/IR data on the HUDF region into the deepest field ever. Ap. J. Suppl. Series 209, [93] Uzan, J.P., Aghanim, N. & Mellier, Y. 6. https://arxiv.org/abs/1305.1931. (2004). Distance duality relation from X-ray and Sunyaev-Zel’dovich observations of clus- [86] Riess, A.G., Filippenko, A.V., Challis, P., ters. Phys. Rev. D 70(8), 083533. https: Clocchiatti, A., Diercks, A., Garnavich, P.M., //arxiv.org/abs/astro-ph/0405620. Gilliland, R.L., Hogan, C.J., Jha, S., Kirsh- ner, R.P. et al. (1998). Observational ev- [94] Nabokov, N.V. & Baryshev, Y.V. (2008). idence from supernovae for an accelerating Classical cosmological tests for galaxies of the universe and a cosmological constant. A. hubble ultra deep field. Astrophysical Bul- J. 116(3), 1009–1038. https://arxiv.org/ letin 63(3), 244–258. https://arxiv.org/ abs/astro-ph/9805201. abs/0901.0405.

[87] Perlmutter, S., Aldering, G., Goldhaber, G., [95] Holanda, R., Goncalves, R. & Alcaniz, J. Knop, R.A., Nugent, P. et al. (1999). Ω and (2012). A test for cosmic distance duality. Λ from 42 high-redshift supernovae. Ap. J. J. Cosmol. Astroph. Phys. 2012(06), 022.

13 [96] Holanda, R., Busti, V. & Alcaniz, J. (2016). [104] Komossa, S. & Hasinger, G. (2002). The Probing the cosmic distance duality with X-ray evolving universe: (ionized) absorp- strong gravitational lensing and supernovae tion and dust, from nearby Seyfert galax- Ia data. J. Cosmol. Astrop. Phys. 2016, 054. ies to high-redshift quasars. arXiv 0207, 321. https://arxiv.org/pdf/astro-ph/ [97] Kauffmann, G., Chariot, S. & White, S.D. 0207321. (1996). Detection of strong evolution in the population of early-type galaxies. Mon. Not. [105] Sethi, G., Dev, A. & Jain, D. (2005). Cosmo- R. Astron. Soc. 283(4), L117–L122. https: logical constraints on a power law universe. //arxiv.org/abs/astro-ph/9605136. Phys. Lett. B 624(3-4), 135–140. http:// arxiv.org/abs/astro-ph/0506255. [98] Van Der Wel, A., Holden, B.P., Zirm, A.W., Franx, M., Rettura, A., Illingworth, G.D. & [106] Jain, D. & Dev, A. (2006). Age of high red- Ford, H.C. (2008). Recent structural evolu- shift objects – a litmus test for the dark en- tion of early-type galaxies: size growth from ergy models. Phys. Lett. B 633(4-5), 436– z=1 to z=0. Ap. J. 688, 48. https://arxiv. 440. https://arxiv.org/abs/astro-ph/ org/abs/0808.0077. 0509212.

[99] Bond, H.E., Nelan, E.P., VandenBerg, D.A., [107] Yang, R.J. & Zhang, S.N. (2010). The age Schaefer, G.H. & Harmer, D. (2013). HD problem in the ΛCDM model. Month. Not. 140283: A star in the solar neighborhood that Roy. Astron. Soc. 407(3), 1835–1841. https: formed shortly after the big bang. Ap. J. let- //arxiv.org/abs/0905.2683. ters 765(1), L12. https://arxiv.org/abs/ 1302.3180. [108] Colafrancesco, S. & Marchegiani, P. (2014). Probing photon decay with the Sunyaev- [100] Creevey, O., Th´evenin, F., Berio, P., Heiter, Zeldovich effect. A&A 562, L2. U., von Braun, K., Mourard, D., Bigot, L., Boyajian, T., Kervella, P., Morel, P. et al. [109] Hughes, J.P. & Birkinshaw, M. (1998). A (2015). Benchmark stars for Gaia Fundamen- measurement of the hubble constant from the tal properties of the Population II star HD X-ray properties and the Sunyaev-Zeldovich 140283 from interferometric, spectroscopic, effect of Cl 0016+16. Ap. J. 501(1), 1. and photometric data. A&A 575, A26. https://arxiv.org/abs/1410.4780. [110] Hawkins, M. (2001). Time dilation and quasar variability. Ap. J. letters 553(2), [101] O’Malley, E.M., Gilligan, C. & Chaboyer, B. L97. https://arxiv.org/abs/astro-ph/ (2017). Absolute ages and distances of 22 0105073. GCs using monte carlo main-sequence fitting. The Astrophysical Journal 838(2), 162. [111] Hawkins, M. (2010). On time dilation in quasar light curves. Mon. Not. Roy. Astron. [102] Fria¸ca,A., Alcaniz, J. & Lima, J. (2005). An Soc. 405(3), 1940–1946. old quasar in a young dark energy-dominated universe ? Month. Not. R. Astron. Soc. [112] Kocevski, D. & Petrosian, V. (2013). On the 362(4), 1295–1300. https://arxiv.org/ lack of time dilation signatures in gamma-ray abs/astro-ph/0504031. burst light curves. Ap. J. 765(2), 116.

[103] Hasinger, G., Schartel, N. & Komossa, S. [113] Littlejohns, O. & Butler, N. (2014). Inves- (2002). Discovery of an ionized Fe K edge tigating signatures of cosmological time dila- in the z= 3.91 broad absorption line quasar tion in duration measures of prompt gamma- APM 08279+ 5255 with XMM-Newton. Ap. ray burst light curves. Mon. Not. R. Astron. J. letters 573(2), L77. https://arxiv.org/ Soc. 444(4), 3948–3960. https://arxiv. pdf/astro-ph/0207005. org/abs/1408.6525.

14 [114] Drell, P.S., Loredo, T.J. & Wasserman, I. [125] Eddington, A.S. (1930). On the instability (2000). Type Ia supernovae, evolution, and of einstein’s spherical world. Mon. Not. R. the cosmological constant. Ap. J. 530, 593. Astron. Soc. 90, 668–678. [115] Kang, Y., Lee, Y.W., Kim, Y.L., Chung, C. [126] Simonsen, J.T. & Hannestad, S. (1999). & Ree, C.H. (2020). Early-type host galax- Can dust segregation mimic a cosmologi- ies of type Ia supernovae. II. Evidence for lu- cal constant? Astron. Astrophys. 351(1), minosity evolution in supernova cosmology. 1–9. https://arxiv.org/abs/astro-ph/ Ap. J. 889(1), 8. https://arxiv.org/abs/ 9909225. 1912.04903. [127] Robaina, A.R. & Cepa, J. (2007). Redshift- [116] Nicolas, N., Rigault, M., Copin, Y., Graziani, distance relations from type Ia supernova R., Aldering, G., Briday, M., Nordin, J., Kim, observations-New constraints on grey dust Y.L. & Perlmutter, S. (2020). Redshift Evo- models. Astron. Astrophys. 464(2), 465–470. lution of the Underlying Type Ia Supernova [128] Hawkins, M.R.S. (2015). A new look at mi- Stretch Distribution. arXiv 2005, 09441. crolensing limits on dark matter in the Galac- https://arxiv.org/abs/2005.09441. tic halo. Astronomy & Astrophysics 575, [117] Crawford, D.F. (2017). A problem with the A107. analysis of type Ia supernovae. Open Astron. [129] Heeck, J. (2013). How stable is the pho- 26(1), 111–119. ton? Phys. rev. lett. 111(2), 021801. https: [118] Bondi, H. & Gold, T. (1948). The steady- //arxiv.org/abs/1304.2821. state theory of the expanding universe. Mon. [130] DeRaad Jr, L.L., Milton, K.A. & Dass, N.H. Not. Roy. Astron. Soc. 108(3), 252–270. (1976). Photon decay into neutrinos in a [119] Hoyle, F. (1948). A new model for the ex- strong magnetic field. Phys. Rev. D 14(12), panding universe. Mon. Not. Roy. Astron. 3326. Soc. 108, 372. [131] Lesgourgues, J. & Pastor, S. (2006). Massive [120] Bondi, H. (1957). Cosmology. Cambridge neutrinos and cosmology. Phys. Rep. 429(6), University Press. 307–379. [121] Narlikar, J.V. (1987). Alternative cosmolo- [132] Bonetti, L., Ellis, J., Mavromatos, N.E., gies. In , pages 447– Sakharov, A.S., Sarkisyan-Grinbaum, E.K. 459. D.Reidel Co. & Spallicci, A.D. (2016). Photon mass lim- its from fast radio bursts. Phys. Lett. B [122] Westphal, T., Hepach, H., Pfaff, J. & As- 757, 548–552. https://arxiv.org/abs/ pelmeyer, M. (2021). Measurement of grav- 1602.09135. itational coupling between millimetre-sized masses. Nature 591(7849), 225–228. [133] Wu, X.F., Zhang, S.B., Gao, H., Wei, J.J., Zou, Y.C., Lei, W.H., Zhang, B., Dai, Z.G. [123] Anderson, J.D., Laing, P.A., Lau, E.L., Liu, & M´esz´aros,P. (2016). Constraints on the A.S., Nieto, M.M. & Turyshev, S.G. (1998). photon mass with fast radio bursts. Ap. J. Indication, from Pioneer 10/11, Galileo, and Letters 822(1), L15. https://arxiv.org/ Ulysses data, of an apparent anomalous, abs/1602.07835. weak, long-range acceleration. Phys. Rev. let. 81(14), 2858–2861. [134] Holanda, R., Silva, K.V. & Busti, V. (2018). X-ray surface brightness observa- [124] Iorio, L. & Giudice, G. (2006). What do tions of galaxy clusters, cosmic opacity and the orbital motions of the outer planets of the limits on the matter density parame- the Solar System tell us about the Pioneer ter. J. Cosmol. Astrop. Phys. 2018(03), 031. anomaly ? New Astronomy 11(8), 600–607. https://arxiv.org/abs/1706.08463. https://arxiv.org/abs/gr-qc/0601055.

15