No obvious change in the number density of galaxies up to z ≈ 3.5 Yves-Henri Sanejouand To cite this version: Yves-Henri Sanejouand. No obvious change in the number density of galaxies up to z ≈ 3.5. 2019. hal-02019920 HAL Id: hal-02019920 https://hal.archives-ouvertes.fr/hal-02019920 Preprint submitted on 14 Feb 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. No obvious change in the number density of galaxies up to z ≈ 3.5 Yves-Henri Sanejouand∗ Facult´edes Sciences et des Techniques, Nantes, France. February 12, 2019 Abstract So, either the methods used for estimating ages of objects like stars or galaxies require significant The analysis of the cumulative count of sources of improvements, or ΛCDM has to be replaced by an- gamma-ray bursts as a function of their redshift other model. strongly suggests that the number density of star- Since ΛCDM is built with still mysterious dom- forming galaxies is roughly constant, up to z ≈ 3.5. inant components like dark energy [13] or non- The analysis of the cumulative count of galaxies baryonic dark matter [14], and since it requires in the Hubble Ultra Deep Field further shows that additional strong assumptions, like an exponential the overall number density of galaxies is constant expansion of space in the early Universe [15, 16], as well, up to z ≈ 2 at least. Since ΛCDM does it may prove worth considering the later hypoth- not seem able to cope with the age of old objects, esis. Hereafter, a non-standard redshift-distance both analyses were performed using a non-standard relationship is thus preferred. Note that it may redshift-distance relationship. serve as an anchor for the development of the next generation of cosmological models. Keywords: Age problem, Milne model, Gamma-ray bursts, Hubble Ultra Deep Field, Galaxy mergers, Splitting events. The age problem Within the frame of a Friedmann-Lemaitre- Introduction Robertson-Walker cosmology for the case of an ho- mogeneous and isotropic Universe, τ(z), the age of ΛCDM, the nowadays standard cosmological the Universe at a given redshift, is so that [13]: model, has proved able to rationalize numerous ob- Z 1 0 servations, of various kinds. However, although the dz τ(z) = 0 0 (1) reintroduction of a cosmological constant, twenty z (1 + z )H(z ) years ago, did help a lot [1, 2], it is still suffer- ing from an age problem [3, 4, 5, 6]. And though with: this problem has been around since the earliest ver- 3 2 1 H(z) = H (Ω (1 + z) + Ω (1 + z) + Ω ) 2 (2) sion of the model [7], the level of accuracy reached 0 m k Λ during the last decade for the measurements of cos- where H is the Hubble constant and where the mological parameters [8, 9, 10, 11, 12] leaves little 0 contribution of Ω , the radiation term, has been room for future major changes of the predicted age r omitted, the radiation-dominated era being much of the Universe. shorter than τ(z) for redshifts considered in the ∗[email protected] present study. 1 Table 1: Estimated age and incubation time of two old, well characterized, objects. The incubation time is defined as the time elapsed between the birth of the Universe, according to ΛCDM or to the −1 −1 Milne cosmological model (with H0=67.4 km s Mpc [12]), and the birth of the object. Negative incubation times are underlined. HD 140283 is an extremely metal-deficient subgiant; APM 08279+5255 is an exceptionally luminous, gravitationaly lensed, quasar. Age Age of Universe Incubation time Object Redshift Ref. (Gyr) (ΛCDM) (Milne) (ΛCDM) (Milne) 14.5 ± 0.8 [17] -0.7 0 14.3 ± 0.8 [18] -0.5 0.2 HD 140283 0 13.8 14.5 13.7 ± 0.7 [19] 0.1 0.8 12.2 ± 0.6a [19] 1.6 2.3 3 [20, 21] -1.4 0 APM 08279+5255 3.9 1.6 3.0 2.1b [22] -0.5 0.9 a With AV =0.1 mag [19]. bThe lowest limit being 1.8 Gyr [22]. ΛCDM The Milne model Let us now consider an open model where Analyses of Planck measurements of the CMB Ωm=ΩΛ=0 (Ωk=1). Thus, eqn (2) becomes: anisotropies are consistent with a flat (Ωk=0) ΛCDM cosmological model, with H =67.4 ± 0.5 0 H(z) = H0(1 + z) km s−1 Mpc−1 and a matter density parameter Ω =0.315 ± 0.007 [12]. m and eqn (1) yields: But with such parameters, according to eqn (1), T ΛCDM can hardly explain how a quasar as old as τ(z) = H (3) APM 08279+5255 can be observed at z=3.9 [20]. 1 + z Indeed, within the frame of the ΛCDM model, this −1 quasar should be at least 0.2 [22] and up to 1.4 Gyr where TH = H0 is the Hubble time. [21] older than the Universe itself (Table 1). This simple, one-parameter model, which is rem- Other objects have been claimed to be older than iniscent of the Milne cosmology [23], belongs to the the age of the Universe predicted by ΛCDM like, in family of power-law cosmological models [24, 25, 26, our neighborhood, the metal-deficient subgiant HD 27, 28, 29]. Interestingly, it has been shown that, 140283 [17, 18]. It has recently been shown that, at least as far as H(z) and τ(z) are concerned, the by assuming an extinction value of 0.1 mag, the predictions of this model are in good agreement estimated age of this star can become comfortably with observational data [25, 26, 27, 30, 31, 32, 33]. lower (12.2 ± 0.6 Gyr [19]; Table 1). However, for Noteworthy, as illustrated in Table 1, it seems able stars as close as HD 140283, interstellar extinction to handle the age problem better than ΛCDM. As a is usually assumed to be non-existent [19]. matter of fact, the Milne model would be seriously challenged only if the upper estimates of the ages of Note that the value of the Hubble constant ob- HD 140283 and APM 08279+5255 were confirmed. tained by the Planck collaboration [12] is signif- icantly lower than values recently obtained using Although a Ωm=0 model is not supported by ob- local measurements [9, 10, 11] meaning that, ac- servational data, note that within the frame of the cording to ΛCDM, the age of the Universe could Dirac-Milne cosmology it is expected to be a fair be as low as 12.7 Gyr [11] (1.4 Gyr at z=3.9). approximation on large scales [28, 34]. 2 Main hypotheses Hereafter, it is assumed that: I. Eqn (3) yields accurate enough predictions for ∆τ = τ(0) − τ(z), that is: z ∆τ = T (4) H 1 + z II. During its travel, a photon ages as the Uni- verse does, namely: ∆t = ∆τ (5) where ∆t is the time taken by a photon to fly from a source at redshift z to an observer on Earth. Figure 1: Cumulative count of galaxies in the Hub- ble Ultra Deep Field (HUDF) and of sources of long III. The speed of light, c0, is constant. gamma-ray bursts (GRB) detected by Swift, as a function of redshift. Both counts were normalized Main consequences using the value expected when z ! 1, according to Hypothesis III yields: a one-parameter function (plain line), as explained in the text. Dc = c0∆t (6) where Dc is the light-travel distance while, with which becomes, with eqn (6) and (7): eqn (5), eqn (4) becomes: z3 z z n(z) = nst 1 + ρ (10) ∆t = T (7) (1 + z)3 1 + z H 1 + z where: Note that this later relationship has been obtained 4 n = πD3 ρ in various contexts [32, 35]. st 3 H 0 and: 3 ρ_ Counts of galaxies ρ = TH (11) 4 ρ0 n(Dc), the cumulative count of galaxies as a func- −1 tion of the light-travel distance, is such that: DH = c0H0 being the Hubble length. Z Dc 2 n(Dc) = 4πρ(r)r dr (8) Datasets 0 where ρ(r) is the number density of galaxies at dis- Studying n(z), that is, a cumulative count of ob- tance r. jects as a function of redshift, requires a fair sam- Let us assume that ρ(∆t), the number density of pling of these objects, for a range of redshifts as galaxies as a function of the photon time-of-flight, large as possible. For this purpose, sources of evolves slowly enough, so that: gamma-ray bursts (GRB) are attractive candidates since their redshifts have been determined up to ρ(∆t) ≈ ρ0 +ρ _∆t (9) z=8.23 [36], while major efforts have been under- taken by follow-up telescopes for determining the whereρ _ is the time derivative of ρ(∆t). With eqn redshift of each of them as accurately as possible (6) and (9), eqn (8) yields: [37]. 4 3 3 ρ_ Dc In spite of this, redshifts are known for only 30% n(Dc) = πDc ρ0 1 + 3 4 ρ0 c0 of the GRBs detected by Swift [38], leaving room 3 for doubts on the fairness of the sampling [39, 40].