Supernovae and their cosmological implications

Supernovae (SNe) are luminous optical transients, which, despite being rare events in a single galaxy, nowadays are discovered at a rate of ten per day. They populate a growing zoo of different types linked to a variety of progenitors and explosion mechanisms. Through a biased selection of a century of researches, I will review how SNe have became a leading tool to probe the cosmic expansion. I will mainly focus on SNe of type Ia, originating from the thermonuclear explosion of white dwarfs, that were demonstrated to be excellent distance indicators after proper calibration and standardisation. Two decades ago, SNe Ia provided the surprising result that the cosmic expansion is accelerated by a misterious dark energy. Recently, with the decisive contribution of Cepheids variables as primary distance indicators, SN Ia allowed to measure the value of the Hubble-Lemaître constant, H0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_0$$\end{document}, with an error of only 1%. It turned out that the value of H0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_0$$\end{document} based on SNe Ia is in tension with the one measured through the fit of the CMB fluctuations. If confirmed, this finding requires a deep revision of the standard cosmological model.


Introduction
The link of supernovae with cosmology began with their first appearance as a special class of astronomical transients. The historical description of the two galactic novae of 1572 and 1604 by Tycho Brahe and Kepler, respectively, showed that these events had to be located beyond the Moon distance because of the lack of measurable parallaxes. This led to challenge the long-standing paradigm of Aristotle's Cosmology that the heaven beyond the Moon was unalterable and nothing new could occur there [1,2].
B Enrico Cappellaro enrico.cappellaro@inaf.it 1 INAF, Osservatorio Astronomico di Padova, Vicolo dell' Osservatorio 5,35122 Padua,Italy In modern times, the interest in novae was reignited by the appearance, in 1885, of a nova in the Andromeda nebula (now named SN 1885A) [3]. This event, along with the novae discovered in the Milky Way and in several spiral nebulae in the following decades, became one of the key topics in the "Great Debate" of 1920 on the nature of spiral nebulae and the size of the Universe [4]. Shapley argued that if the nebulae were isolated distant galaxies their novae would need to have "impossibly bright" absolute magnitudes. 1 On the other hand, Curtis noted the large dispersion of magnitudes of galactic novae and suggested that "a division into two classes is not impossible", a conclusion that was later reinforced by Lundmark [5]. Eventually, with the crucial contribution of the discovery of Cepheids variables in several nebulae [6], they were recognised as extragalactic stellar systems. Hereafter, Baade and Zwicky [7] coined the term super-nova for the class of intrinsically bright novae with absolute magnitudes comparable to those of the entire stellar system where they appear.
The first systematic search for SNe was conducted in the late '30s by Zwicky with a dedicated wide-field telescope located at Palomar Observatory. It leads to the discovery of a dozen SNe and to estimate their rate of about 1 SN per nebula over 400 years [8,9].
Minkowski [10] first noticed that SNe were of at least two different types, with the type II showing the hydrogen Balmer in their spectra, while this feature is not seen in type I. This was the beginning of a long journey that in the following decades had many new branches and surprises and it is not yet concluded [11].
After the first Zwicky's search and a few years with only casual discoveries, the search for SNe got a new impulse in the late '50s with the coordinated effort of the newly available wide-field Schmidt telescopes, in particular, the Palomar 48inch (USA) but also the telescopes in Zimmerwald (Zwitzerland), Asiago (Italy) and Tonantzintla (Mexico) [12,13]. The effort was very successful, and several hundred SNe were discovered until its completion in the mid '70s.
In the '80s the SN searches were continuing at a reduced pace with the discovery of only a dozen SNe per year until the next milestone, the discovery of SN 1987A in the Large Magellanic Cloud, the first SN visible to the naked eye since 1604. The overwhelming amount and quality of observations of this event provided unique tests for long debated issues in stellar evolution and core collapse (CC) theories [14] and gave a great boost to SN research that became a hot topic in astrophysics: nowadays searches produce thousands SNe per year (source for the transients statistics is the Transient Name Server, hereafter TNS, https://www.wis-tns.org).
In this review, after an overview of the current SN taxonomy (Sect. 2) and the progresses of SN surveys (Sect. 3), I will focus on the use of SNe as cosmological probes. In particular, I will describe the use of SNe as tracers of the cosmic star 1 The 'apparent magnitude' is a measure of the luminosity of an astronomical source on an inverse logarithmic magnitude scale. Bright sources have small, or even negative magnitudes. As a reference, the apparent magnitude of the Sun is −26.7 while the faintest stars visible to the naked eye have a magnitude about 6. Instead, the absolute magnitude is a measure of the intrinsic luminosity of a source and is related to the apparent magnitude through the source distance via the relation M = m − 5 log 10 D L 10 pc . Th absolute magnitude of the Sun is 4.8 mag, that of the whole Andromeda galaxy is −21.5 mag. D L , the 'luminosity distance', depends on the cosmological model and, therefore, the measurement of apparent magnitudes for sources of known absolute magnitudes allow for cosmological tests.  6,7), and of the cosmic expansion history (Sect. 8). Finally, I will present the current standing of the debate on the so-called Hubble tension that arises from the discrepancy of the Hubble constant measured by different probes, in particular Cepheids plus SNe Ia, and the value derived from the fit of the cosmic micro-wave background (CMB) (Sect. 9).

Supernova types
The taxonomy of SNe relies mainly on spectroscopy. The top-level branch of the nomenclature separates events showing H features in the spectra (type II SN or SN II) from those that do not show them (SN I). Most SNe I show a broad absorption feature at ∼ 615 nm that is attributed to the SiII 6355 Å doublet. 3 Some "peculiar" SNe I missing this feature was noted early on [15] but only in the mid '80s SN I were divided in type Ia and Ib based on the strength of this spectral line [16] but also on distinctive infrared features [17]. In particular, the near-infrared (NIR) light curves of type Ia show a second maximum about one month after the first peak that is not seen in type Ib. Then, the spectra of some SNe Ib show the presence of strong He lines (in particular the HeI 5876 Å) while other SNe I show neither strong Si nor He features but often a strong OI 7774 Å line. The latter events were labelled SN Ic [18].
Conventionally, the SN classification is based on spectra obtained near maximum light (Fig. 1, left panel). However, the SN types become more diverse after the comparison of the spectra obtained at a late time, i.e. few months after the explosion in the so-called nebular phase, when the ejecta become transparent and the ashes of nuclear burning and explosion are revealed. As shown in Fig. 1 (right panel), at late phases the spectra of SNe Ib and Ic are very similar, both dominated by O and Ca features and also similar to SN II but for the Hα emission. SN Ia spectra are completely different being dominated by strong Fe and Co features. It is now accepted that all SN II, Ib and Ic originate from the core collapse of massive stars. After exhaustion of the nuclear burning, massive stars are left with an onion-like structure, with progressively lighter elements moving from the center to the stellar surface. The collapse of the Fe core causes the ejection of the star envelope, which outermost layer can have different composition, mainly H in type II, He in type Ib, and deprived of both H and He in type Ic. The difference in the envelope composition occurs because of variable mass loss that, in the most extreme cases, can be also related to close binary evolution [19].
There are two issues complicating the assignment of a SN type, even for regular events. On one hand, the SN spectra rapidly change with time and they can mimic different types at different times. For instance, the spectra of SN Ia two weeks after maximum almost match those of SN Ic at maximum and the two types can be confused when a light curve is not available. On the other hand, there are true transitional cases where a distinctive feature is present but weak, eg. some SN Ic showing faint He lines [20]. The two issues can occur in combination: there are events showing H features early-on that then disappear, while He features emerge. These SNe are thought to result from massive stars that have lost most, but not all, their H envelope. They are genuine intermediate cases between type II and Ib and are labelled SN IIb [21].
The different spectroscopic SN types also show different luminosity evolution, that is different light curves. By itself, the explosion of the SN would provide only a rapid transient event (lasting a few days) because adiabatic losses in the fast-expanding ejecta would cause rapid cooling and dimming. However, nucleo-syntesis during the SN explosion produces radioactive species (in particular 56 Ni) whose decay deposit energy in the ejecta. In most SN types, this source is powering the long lasting luminosity.
With regards to the luminosity evolution, SNe Ia show the highest homogeneity among SN types. Actually, it is recognised that, strictly speaking, even SNe Ia are not standard candles since they show significant diversity in their absolute magnitudes at maximum. Standardisation methods have been developed promoting their use as powerful cosmic distance indicators (cf. Sect. 6).
Type II SNe, on the other hand, span a wide range of properties. A first distinction was introduced between SNe II with a flat light curve during the early 2-3 months after explosion (type II Plateau or type IIP) and those showing a monotonic decline (type II Linear or IIL) [22]. The two classes are not really separated but most likely there is a continuous distribution of light curve shapes [23][24][25]. This distribution extended beyond the original definition with the discovery of the very nearby SN 1987A, then becoming the prototype of SNe II with a long rise to a delayed maximum (2.5 months) rather than a plateau.
The variety of SN II light curve shapes goes with a wide range of absolute luminosities at peak extending from −14 to −19 mag [26] and, therefore, one may expect that SNe II are not useful distance indicators. However, methods have been elaborated for the standardisation of the selected sample of SNe IIP that can provide reliable distance estimates (cf. Sect. 7). Spectroscopically, except for the ubiquitous presence of Hα, SNe II show significant differences in line strengths and profiles. Most SNe II near explosion show a blue, almost featureless continuum. Then, in a few days they develop H Balmer lines with a broad emission at the rest-frame transition wavelength and a blue-shifted, broad absorption (P-Cygni profile). The relative strength of the absorption/emission components can be very different and it seems to correlated with the SN luminosity [27].
A fraction of SNe II shows narrow emission profiles of the H Balmer lines, often on top of broader components [28]. It is now understood that the narrow lines originate in circumstellar material (CSM) surrounding the expanding SN ejecta and excited by the flash of radiation emitted when the shock wave, originating in the SN explosion, emerges at the stellar surface (shock break-out). The ejecta of the SN, moving at high velocity, eventually collide into the slow-moving CSM. The timing and outcomes of the ejecta-CSM shock depend on the radius and density of the CSM shell and on the geometry of the system, in particular on the presence of asymmetries. Variations in these ingredients can cause very different transient phenomena: from peripheral, though spectacular, effects such as the rings of SN 1987A [29], to the prolonged emission of the otherwise normal SN IIL 1979C [30], or even, fully dominating the SN display, slowly evolving, high luminosity type IIn, such as SN 1988Z [31], often also sources of strong radio and X-ray emission. In some cases, the spectrum shows narrow lines of He rather than H (SN Ibn) that is thought to reflect a He-rich composition of the CSM [32].
A dense CSM may result from strong winds or from outburst(s) in very massive stars but may also occur in moderate massive stars as a consequence of close binary evolution. Actually, a non-terminal outburst of a massive star can shock pre-existing CSM, produced from stellar wind or previous outbursts, and give rise to a bright transient, mimicking an SN event. These so-called SN impostors can be very difficult to properly recognise because the radiation from ejecta/CSM shock hide the innermost engine, either an erupting massive star or a real SN [33]. Occasionally, transients with the spectral appearance of type IIn were actually found to be SN Ia that were embedded in a dense CSM, possibly the relic of a common envelope ejection [34,35]. Except for these relatively rare cases, SNe IIn are typically associated with massive stars. A text-book case is that of SN 2009ip. Discovered after an outburst in 2009, not the first one for this star [36], SN 2009ip eventually exploded in 2012 [37] but the ashes from the core collapse remain hidden because of the curtain of the ejecta/CSM shock [38]. With the improved depth and sky coverage of modern transient surveys, an increasing number of cases similar to SN 2009ip are being identified [39].
A sub-class of SN Ic that is receiving special attention includes the so-called hypernovae. The prototype of these objects is SN 1998bw found during the follow-up of the gamma-ray burst GRB 980425 [40]. The spectrum of SN 1998bw is of a type Ic but, because of a very high expansion velocity of the ejecta, the lines are broad (these events are also labelled SN IcBL, where BL stands for broad line). The energetic explosion is thought to originate from a massive star stripped of the H and He envelope that collapses into a black hole. Possibly due to a high rotation of the progenitor star, the ejecta are highly asymmetric and the collapse powers relativistic jets that manifest with a burst of γ -ray emission [41]. The GRB emission is strongly beamed and there-fore can be seen only if the jet is aligned with the line of sight. On the other hand, the thermal radiation from the SN ejecta is emitted isotropically. This would explain why there are SNe IcBL for which a GRB was not detected.
The SN types considered so far have a broad luminosity distribution with an upper limit at an absolute magnitude of about −20 mag. This limit was broken with the discovery of super-luminous supernovae (SLSN) [42]. SLSNe have their own subclass branching with type I and type II, depending on the absence/presence of H lines. Their distinctive feature is that their absolute magnitude is well above −21 and up to −23 mag and, in general, show slow-evolving light curves. The rise to maximum occurs with a time scale of 1-2 months rather than the 1-2 weeks of ordinary SNe and the same applies to the luminosity decline [43]. The spectra at early phase show a very blue continuum that in SLSN II is almost featureless [44] while in SLSN I is characterised by broad absorption lines of OII at 350-450 nm [42]. At late phases, SLSNe II develops broad H Balmer lines while the spectra of SLSNe I resemble regular SNe Ic [45]. While it is widely believed that the progenitors of SLSNe are very massive stars, the mechanism powering the very high luminosity is not yet fully understood. There are two prevalent alternatives: (i) deposition in the ejecta of the spin-down rotation energy from a newly born, highly magnetic neutron star (magnetar) or (ii) the formation of a shock at the interface of the ejecta with a very dense CSM. In some cases, the contribution of both mechanisms seems to be required. The production of a very large mass of radioactive material in a pair production SN seems not favoured [46].
SLSNe are rare, one out of a thousand regular SNe, but because of the enormous luminosity they can be detected up to 10 billion light years. Therefore, assuming that they can be standardized, they can be used as cosmic probes at redshift z > 1 ÷ 2 (Sect. 7).

Modern supernova searches
The intensity and productivity of SN searches depend on the strengths and weaknesses of the dedicated efforts but also on the instrument capabilities. A technical milestone was, in the early '80s, the introduction in astronomy of CCD detectors that suddenly augmented the performances of all existing telescopes. At the beginning, however, the gain in limiting magnitude was at the expense of the small size of the detector, and then a drastically reduced field of view (FoV), from the few degrees of the photographic plates at Schmidt telescopes to the few arcminutes of the new devices. In the struggle between opportunities and limitations, the SN searches took two directions: from one hand, there were the first attempts to implement automatic SN searches on nearby galaxies, exploiting the digitally recorded images [47]. On the other hand, the deeper limiting magnitude allowed to increase the search distance and to start high redshift (z > 0.2 ÷ 0.3) SN searches [48].
In the early 2000s, the monitoring of thousand galaxies per night with robotic telescopes increased the SN discovery rate up to a few hundreds events per year. It is important to recognise that all SN searches have biases dictated by technical limitation but can also be structured (e.g. cadence and field selection) to promote the discovery   [49]) were preferentially monitoring nearby giant spiral galaxies that were known to be more productive than other galaxy types. At the other extreme, high redshift SN searches were planned with a cadence of observations tuned to the time-scale of SN Ia, for they intended to use as distance scale indicators [50].
Eventually, the introduction of CCD mosaic detectors (e.g. [51]) allowed to recover the full telescope's FoV, making again possible the search of large sky areas, with no apriori selection of target galaxies. This is one reason for the relatively recent discovery of SLSNe that have a preference for intrinsically faint host galaxies penalised in galaxy targeted searches. The other key advancement came from a wider exploration of the time domain. The accumulation of rich archive material for wide sky areas allowed the search for transients with long term temporal evolution, again SLSNe, but also the recovery of pre-explosion outbursts for regular SNe. On the other hand the increased survey efficiency allowed to extend the exploration to the short time domain, with the discovery of fast peculiar transients (see SN 2018cow for a representative case [52]).
Overall, the SN discoveries in the last 5 years largely outnumbered those of the previous century. Currently, almost 95% of the confirmed SNe are discovered by five major surveys (Fig. 2) with the major player being ZFT delivering half of the SN discoveries. The number of candidate SN discovered by modern transient surveys is so large, almost a hundred per day, that only a small fraction (∼ 10%) receives a spectroscopic confirmation (source TNS).
The transient detection rate is expected to dramatically increase with the coming into operation of the Vera Rubin Telescope that is predicted to discover millions of SN candidates during the decade of the Legacy Survey of Space and Time (LSST). 4 However, we cannot expect the number of spectroscopically confirmed SNe to grow proportionally. As I mentioned above, even at the current detection rate, candidate confirmation is limited by the availability of spectroscopic classification facilities that are not expected to grow at the same rate as transient discoveries. To make things more . Both images were obtained with the Asiago 67/91 Schmidt Telescope. The distance of the host galaxy is only ∼ 6.5Mpc that makes of SN 2011fe the fourth closer (extragalactic) SN Ia ever discovered difficult, LSST will explore a much bigger volume of the Universe than current surveys, hence the candidate apparent magnitude distribution will extend to a much fainter limit. This will require longer exposure time per target or wide access to over-subscribed large telescopes. To deal with this oversubscription of spectroscopic facilities, the community is working to develop methods for photometric classification of transients, for an efficient selection of targets for spectroscopic follow up but, also, to increase the statistics of SN samples for specific science goal.
However, it is not expected that, with the new surveys, the number of local SNe, the key for the distance scale calibration, will grow rapidly (cf. Sect. 5). In fact, the number of SN discoveries in the second decade of this century is almost three times larger than that of the first decade, yet the number of nearby events within 15 Mpc is almost the same. This suggests that the transient surveys in the local Universe are close to being complete, at least for bright transients such as SNe (Fig. 3).

Progenitor scenarios and the cosmic star formation history
The statistics of SNe of different types can be used to constrain the progenitor scenarios, when we know the properties of their host stellar systems. Conversely, when the SN progenitor evolutionary paths are known, SN rates can be used to probe some of the key ingredients for the evolution of the Universe, i.e. the star formation history (SFH) and the initial mass function (IMF). SN rates are also key inputs for the models of galaxy evolution because they regulate the chemical evolution and the stellar feedback. Stellar evolution theory predicts the observed properties of stars during nuclear burning and their fate after exhaustion of the nuclear fuels. The main parameter determining a star's evolution is its initial mass, M 0 , then come metallicity, rotation, mass loss, close binary interaction, etc. It is found that the larger the mass, the shorter the star lifetime is, hence the more massive stars are found in stellar populations with younger ages. With this in mind, a first clue for the progenitor scenarios came from the fact that SN II occur only in galaxies with a young stellar population, while SN I occur in all types of galaxy [53]. This immediately suggests that progenitors of SN II are more massive than those of type I. In reality, SN I also are more frequent in star forming spiral galaxies than in passive elliptical ones. Early statistics did not distinguish between SN Ia and SN Ib/c, with the latter only occurring in star-forming spiral galaxies [54], but even the rate of SN Ia alone is higher in galaxies with recent star formation [55], especially after normalisation to the galaxy mass [56]. This requires that a significant fraction of SN Ia events in late spirals/irregulars originates in a relatively young stellar component and are associated with moderately massive progenitors.
Combining these observations with the prediction from stellar evolution models, SN scenarios were elaborated with the main discriminant being the progenitor star mass. Stars with masses above 8 − 9M complete all the core nuclear burning until they build an onion-like structure, with an iron core and progressively lighter elements in the outer shells. When nuclear reactions cease to support the star, gravity causes the collapse. The rise in temperature in the core leads to photo-dissociation of the iron nuclei into alpha particles and the collapse proceeded at an even higher pace. At high density, electrons and protons collapse to neutrons, and a huge neutrino flux is emitted. A small fraction of the neutrino flux is transferred to the stellar envelope that is then expelled at high velocity, although the exact mechanism causing the envelope ejection is still the key question for SN explosion theories [57]. Depending on the amount of H left in the star envelope at the time of explosion, the outcome may be a type II or type Ib/c SN, respectively. At the center of the explosion remains a compact remnant, a neutron star (NS) or, if due to fall-back of material, the NS overcomes the stability limit, a black hole (BH). For mass above 25 ÷ 40M , the whole star collapses into a BH and no SN is produced (failed SN or dark collapse). The exact mass limit is uncertain and depends on other parameters, such as the star rotation speed [58]. Finally, in stars with mass above ∼ 100 ÷ 140M , after the ignition of carbon nuclear burning, electron-positron pair production reduces the gas pressure and causes the collapse. Ignition of the nuclear fuels leads to a powerful explosion that completely destroys the star leaving no compact remnant. It has been argued that the peculiar characteristics of SN 2007bi can be attributed to this type of explosion [59], though the evidence is still not conclusive.
Stars with masses below 8-9 M do not reach the temperature needed to ignite C fusion in the core. Eventually, these stars become C/O white dwarfs, compact objects sustained by electron-degenerate gas with a mass less than the Chandrasekhar limit (M CH ∼ 1.4M ). The fate of an isolated WD is to remain stable, while slowly cooling. However, a WD in a close binary system may accrete matter from the companion and, in special conditions, reach the M CH limit. When this happens, the star is destroyed by an explosive thermonuclear burning that produces iron-peak elements. The outcome of this explosion is a type Ia SN. Two main alternatives have been proposed for the progenitor system [60]. In one case the companion of the WD is a main sequence, or a red giant star, that transfers mass through Roche lobe overflow (single degenerate system or SD). The other case is a system made of two WDs that merge after loosing  [67,68]. The SFH from literature adopted different reference IMF hence require a different scaling factor. Right panel: rates of SN Ia as a function of redshift are compared with a prediction for different delay time distribution. Updated from [67] orbital energy, due to gravitational wave emission (double degenerate system, DD). These two channels may actually coexist and partially explain the SN Ia diversity. It has been argued that a SNe Ia can occur even for WD masses below the M CH limit, when the detonation of an accreted layer of H or He ignites a subsequent explosion of the C-O WD [61]. In contrast with earlier claims, recent simulations suggested that this sub-Chandra scenario produce photometric and spectroscopic properties in good agreement with those of SNe Ia [62]. Whether this channel can explain the majority of SNe Ia is still unclear, but it seems required at least for the production of faint, fast-declining events [63].
In all cases, the clock is set by the evolutionary time scale of the less massive star in the system. After that, one needs to consider either, the mass transfer and accretion rate for SD systems or, the orbital shrinking rate by gravitational wave emission for DD systems [64]. In both cases, binary evolutionary models can predict the time span from formation to explosion depending on the initial star masses and on the system orbital parameters. This means that, for an adopted distribution of the system initial parameters, it is possible to derive the delay time distribution (DDT) corresponding to the different types of progenitor scenarios. Convolving the DDT with the cosmic SFH, one can predict the evolution of the SN Ia rates with redshift. The normalisation factor (k Ia ) is the product of the fraction of stars in the useful mass range for given stellar population times their fraction in suitable, close binary systems.
Core-collapse is the final event in the life of massive stars (M > 8 M .) The lifetime of a star during the nuclear burning phase is, to a good approximation, the time spent in the H core burning phase. This time scales with the stellar mass as t 10 10 (M/M ) −2.5 year and, therefore, for stars in the range 8 ÷ 40 M , the time from formation to SN explosion is between 50 and 1 Myr. This time delay from star formation to explosion is short compared with the timescale of the cosmic star formation history, hence the rate of CC SNe can be taken to scale directly with the star formation rate at that redshift. The scaling factor is readily derived knowing the shape of the IMF and the progenitor mass range: eg. for a Kroupa IMF [69] ∼ 1% of stars are born with mass in the range 8 ÷ 40 M . The number decreases by 30% if the lower limit is raised to 10 M .
In the left panel of Fig. 4, the evolution of the rate of CC SNe as a function of redshift is compared with the expectation from two different estimates of the SFH [65,66] and a reference progenitor mass range of 8 ÷ 40M . The fact that the predicted rate from [65] (HB06) is significantly higher (by a factor ∼ 2) than the observed value is still a debated issue in the literature. The discrepancy would be reduced by rising the lower limit of the SN progenitor mass range to 10 ÷ 40M (changing the upper limit would have a small impact). However, the direct detection of red giant progenitors for some SN II seems to tightly constrain the minimum mass to 8 ± 1M [70]. This motivated the claim that the rate of CC SN is under-estimated because a significant fraction of the CC SNe are missed in SN searches being very faint, either intrinsically or due to dust obscuration [71]. In particular, dust obscuration is expected to be very high in the nuclei of luminous infrared galaxies that dominate the cosmic star formation at redshift z > 1 ÷ 2 [72]. I notice however that the prediction from a different SFH estimate (MD14 [66]) is in excellent agreement with the rate measurements and hence, the uncertainty in the SFH calibration should not be overlooked.
The fact that SN Ia explode preferentially in star-forming galaxies but occur also in passive elliptical galaxies where star formation ceased billion years ago, indicate that SN Ia progenitors are moderate/low mass stars (3 ÷ 8M ) in systems that allow for a wide range of evolutionary times. WDs in close binary systems fit this requirement. As I mention above, there are different possible configurations for the binary system. Adopting an SD scenario, the comparison with the observations suggests a value of k I a ∼ 10 −3 , in tension with the prediction of binary population synthesis (BPS) models that typically indicates a normalisation value smaller by a factor 3-4 [73]. The tension is alleviated if we accept that different evolutionary channels can lead to SN Ia explosions, SD and DD, including also sub-Chandrasekhar WDs.

SNe as distance indicators
A luminosity distance indicator is an astrophysical source that is easily recognised as a member of a class, is bright enough to reach useful distances and, occurs with a sufficiently high rate to allow the collection of fair statistical samples. The putative distance indicators need to meet a number of requirements, namely: (i) they reach all the same absolute magnitude (standard candles) or can be standardised to a reference value, based on some distance-independent observed properties, (ii) they can be properly calibrated and (iii) extinction corrected. In addition, they should suffer little contamination from "impostors" and the physics that determine their brightness should be relatively well understood. Each of the items above must be carefully considered because it identifies a source of uncertainty that, eventually, set the limit on the precision of the distance indicator.
Among SN types, SN Ia are the most popular luminosity distance indicators on a cosmological scale. As we will see (Sect. 7), there have been promising attempts to use SNe IIP, that so far remain confined to some demonstrative applications and, more recently, SLSNe that, in principle, can allow to reach much larger distances.
Because SNe are relatively rare, they are not available to measure the distance of a pre-selected galaxy, but they are best used to measure global properties of the Universe, in particular, the cosmological expansion rate, H 0 , and its evolution with cosmic time. The measurement of the evolution of the cosmic expansion rate can be done with SN Ia alone, by comparing the magnitude of local and distant SNe in the Hubble flow. 5 Instead, for the measure of H 0 we need to secure the SN Ia calibration by means of a primary distance indicators, for instance Cepheids variables, that can be observed both in the Galaxy and in nearby SN host galaxies. In turn, the primary distance indicators are calibrated by geometrical methods, parallaxes in particular, that can achieve great accuracy but only at relatively short distances [74].
Geometrical, primary and then secondary distance indicators are the steps of what is called the astronomical distance ladder. In the modern version of the distance ladder, the struggle is to reduce the steps and the number of independent distance indicators, with the aim to reduce the dispersion of measurements and the associated statistical error. On the other hand, it is also important that alternative ladders' steps are explored to check for possible systematics. For the link between the primary local calibrators and the SNe in the Hubble flow since the early days the first option was Cepheid variables, with the most often considered alternatives being the Tully-Fisher relation (TFR), the galaxy surface brightness fluctuations (SBF) and the tip of the red giant branch (TRGB) (c.f. Sect. 5.4).
In this review, I will revisit the history of the use of SNe Ia as distance indicators. I will argue that, in this context, SN Ia lead to important discoveries that, in turn, raised new questions, in particular for the cosmological model. This, from one hand motivated new efforts in the analysis of possible sources of systematics and on the other hand, the quest for more precision, that is requiring more elaborated sample selection and data treatment.

SN photometry
It is manifest that to use SNe as luminosity distance indicators, the first requirement is accurate, properly calibrated photometry.
SNe are embedded in the diffuse light of their host galaxies, that needs to be properly subtracted for accurate photometry. The target may suffer contaminations from nearby sources, foreground stars, compact clusters or HII regions in the host galaxy. If the instrument allows to resolve the contaminating sources, point spread function photometry permits us to make reliable measurements. Otherwise, an alternative approach for accurate photometry, is to subtract an image of the same sky area where the SN is not visible, that is obtained before or long after the SN explosion. Both methods strongly benefit from the introduction in the mid '80 of the linear CCD detectors and, therefore, modern analyses usually neglect the SNe with only photographic plate photometry.
The calibration of instrumental magnitudes can also be tricky. The response curve of a photometric system depends on the transmission of the optics, the filters, and the sensitivity curve of the detector. In fact, virtually none of the existing observing facilities strictly follows a photometric standard definition. SN photometry is especially sensitive to these deviations because the SN spectrum is not a smooth continuum but shows strong absorption and emission features that change with time. It is found that individual instrumental peculiarities introduce systematic deviations in the measured photometry of the same target observed with different telescopes. To cope with this problem, one may choose to limit the comparison only to SNe observed with a single instrument that it is feasible using a dedicated telescope to observe SNe in the Hubble flow. Observing target galaxies in the Hubble flow accesses a fair number of SNe in a program of a few years [75].
On the other hand, the instrument-specific deviations can be corrected if the instrument passbands are known with sufficient accuracy and adequate spectroscopic monitoring of the target is available (often referred to as S-correction [76]) although, securing accurate measurement of the telescope/instrument/detector throughput, may require dedicated calibration programs [77]. In fact, the quest for better precision of SN photometry motivated new approaches for photometric calibration relying on laboratory flux standards (eg. calibrated photodiodes) instead of a stellar standard reference. This allows to improve the calibration accuracy from the 1-2% achievable with photometric standards to 0.2-0.5% [78].
Even when the SNe are observed with the same instrument, for the photometric comparison we need to correct for systematics introduced by their different redshifts (K-correction) [79]. This can be derived by computing the difference of the SN synthetic photometry obtained after convolution of the photometric system transmission curve with the source spectrum as seen in the observer and in the rest frame. Unfortunately, obtaining an adequate spectral sequence for high redshift SNe is often not possible. The typical approach for the estimate of the K-correction exploits the spectral sequence of well observed template SNe that are simulated as observed at different redshifts and also with variable amount of extinction. The latter affects the observed spectral energy distribution and hence the value of the K-correction, even for SN Ia at the same redshift. While the application of the K-correction is crucial for the use of SNe as distance indicator, the correction uncertainty is relatively small: at low redshift the K-correction is intrinsically small, while at high redshift the error budget is rather dominated by the uncertainties in the multi-colour photometry [79].

SN Ia diversity
The use of SNe Ia as distance indicators relies on their great homogeneity, which is much better than other SN types. Actually, whether SNe Ia should be considered "standard candles", all with the same absolute magnitude at maximum, or rather they have a significant luminosity dispersion, was debated for many years (see [80] for the detailed story). As for many other topics in astrophysics, the progress was driven by two components: better event statistics and more accurate photometric measurements.
Already in the early '70s, it was claimed that the SN Ia peak luminosity is brighter for those events with faster luminosity decline, as measured in the early post maximum phase [81][82][83]. However, at that time the number of SN host galaxies with reliable distances was limited and researchers were forced to combine different, not homogenous, methods to estimate galaxy distances including Cepheids, novae, HII regions, brightest stars, cluster memberships along with redshifts (e.g. [84]). The distance modulus uncertainties combined with the poor accuracy of the SN photographic photometry, including possible contamination from the host galaxy [85], explain why these results were disputed for many years. Even as late as the early '90s, in the literature there were claims that the observed dispersion of SN Ia absolute magnitudes was not intrinsic but, rather, entirely due to photometric errors [86,87].
A breakthrough occurred in 1991 with the discovery of two extreme events, namely SN 1991T, a slow-decline luminous SN Ia, and SN 1991bg, a fast-decline faint SN Ia, the former being over 2 mag brighter than the latter (see Fig. 6 left panel for an illustration of the differences among SN Ia light curves). At this point, Phillips [88] assembled a sample of SNe Ia selected for their accurate photometry and accurate relative distances, derived from only two, redshift independent, methods, i.e. SBF and TFR. Despite the relatively small sample (only 9 SNe), it was finally established the existence of a correlation between SN Ia absolute magnitude, M, and their luminosity evolution, the latter characterised by Δm 15 , the difference between the magnitude at maximum and that 15 days later (often called Phillips relation).
Yet, the accuracy of the standardisation of SN Ia absolute magnitudes was limited by the uncertainty of the distance estimates for the still small SN sample. A key contribution came from a dedicated observing program designed to secure accurate photometry for a sample of SNe Ia in the Hubble flow [89]. The richer SN sample allowed a better constrain of the slope of the Phillip's relation [90,91] although the zero point calibration still relied on a small sample of nearby galaxies with distances from SBF and Cepheids (now neglecting the scattered TFR estimates).
In addition to searching for better absolute calibration of the individual SNe, different metrics to characterise the SN photometric evolution were introduced. Among these, the "stretch factor", a parameter used to expand, or contract, linearly the timescale of a particular light curve to match a template [92]. Other studies pointed to a correlation between the SN absolute magnitudes and their intrinsic colours, either measured at some specific phase (i.e. 12 days after explosion [93]) or along the luminosity evolution (i.e. through a "colour-stretch" parameter [94]) A critical step for the determination of the SN Ia absolute magnitude is the extinction correction, in particular, that occurring in the SN host galaxies. As I will discuss later, SN Ia extinction is usually estimated by comparing the observed colour for a given object with that of extinction-free templates. The problem is complicated by the fact that SNe Ia with different luminosity evolution also show a different intrinsic colour. This led to the development of methods performing a simultaneous multicolour comparison to a library of template SNe Ia and incorporating the extinction correction in the light curve parametrisation (e.g. MLCS [95]) or even, in the more recent implementations, account also for the K-correction (MLCS2k2 [96], SALT2 [97], SiFTO [98]).  [75]. Host galaxies distances are derived from the redshifts but for z < 0.01 for which I used redshift-independent distances from NED. The line shows the Phillips relation [91] It has been shown that, allowing for the pros and cons of each approach and restricting the allowed range of decline rates, there is a monotonic correlation between the different parameters used for the standardization of SNe Ia [99].
An example of a typical plot of absolute magnitudes versus luminosity decline rates is shown in Fig. 5 for a sample of SNe Ia taken from [75]. The dispersion of the points appears consistent with the errors but for the fast-decline SNe Ia (Δm 15 (B) > 1.5) that, in some cases, show significant deviation from the adopted Phillips relation. Interestingly, the light curves of these faint events have a distinctive feature, i.e. they lack a secondary maximum in red/near-IR bands that is present in normal SNe Ia.
The essence of the SN Ia standardisation methods is illustrated in Fig. 6, using the stretch-factor approach. The left panel shows the absolute V light curves for a (small) sample of SNe Ia with different decline rates. In the right panel, the same light curves are shown after (i) application of the individual time scale stretching factor and (ii) scaling of the absolute magnitudes based on the stretch-magnitude relation. The "collapse" of light curves on a single standardised template is the key to the confidence on SN Ia as accurate distance indicators.
I emphasize that Figs. 5 and 6 includes only "normal" SNe Ia. An extended version of this plot including all kind of events that have been classified as "thermo-nuclear transients" is shown in [101] (their Fig. 1). This extended sample includes events with "peculiar" spectroscopic or photometric features. Among these are SNe Iax, which spectrum is similar to that of normal SNe Ia at the time of maximum but for narrower lines, signature of a much lower expansion velocity. In addition, SNe Iax never shows the transition to the typical nebular phase [102]. Other SNe Ia, labeled 02es-like from their prototype, have fairly regular spectra but they do not conform to the Phillips relation, appearing significantly sub-luminous for their slow luminosity decline. At the bright end of the luminosity distribution, the majority of SNe Ia seems to obey the Phillips relation (SN 1991T-like), but for a few SNe Ia that are significantly brighter than expected from their decline rate. These SNe Ia seem to have ejecta masses larger than the conventional 1.4 M WD and are sometimes referred to as "super-Chandrasekhar" [101]. These super-Chandra SN Ia seem to be intrinsically rare (∼ 1% of SN Ia) and hence are not expected to produce significant contamination in the SN Ia cosmological sample. At the bright end of the SN Ia luminosity function Fig. 6 Illustration of the standardization of SN Ia light curves using the stretch factor. Left panel shows the light curve in absolute V magnitude (corrected for extinction) for a sample on nearby SNe Ia with different decline rates. Right panel: after stretching the time axis to match the luminosity evolution, the luminosity is scaled based on the light curve evolution-luminosity relation. Data from [100] is located also SN Ia-CSM, in which the type Ia features are hidden behind the curtain produced by the interaction of the ejecta with a dense CSM (cf. Sect. 2). Picking up these events from the more common SN IIn requires carefully scrutiny of high S/N spectra. Given their unique observed features, again, there they are not considered problematic for the cosmological SN Ia sample. The same applies to the deviation at the faint end of the Phillips relation because it is argued that only a few faint SNe Ia enter in the cosmological sample, being strongly biased against in magnitude limited surveys. All these assumptions have to be constantly verified because, if they fail, they can introduce distance-dependent biases.
An important hint for a physical explanation of the observed diversity of "normal" SNe Ia, is that some spectroscopic features correlate with the decline rate [103]. It is now recognised that the SN luminosity and the decline rate correlate with the mass of radioactive 56 Ni in the ejecta and, for normal SN Ia, ranges from 0.1 to 1.0 M [104,105]. There is also a correlation between the SN Ia photometric class and the host galaxy property with less luminous, fast-declining SNe Ia being found preferentially in old elliptical galaxies while bright, slow-declining events occur mainly in star forming, spiral galaxies [90]. This suggests that the SN photometric class, hence the 56 Ni production, depends on the age of the progenitor systems. It is also reported that the most luminous SNe Ia are produced in the most luminous galaxies and metal-poor neighbourhoods [106] and therefore, metallicity could also play a role in determining the outcome of the explosion. Despite the circumstantial evidence, most researchers would agree that we still do not fully understand yet why SN Ia explosions produce such a sequence of 56 Ni masses and we still need to rely on empirical findings.
With respect to the use of the luminosity vs. decline rate relation, the discussion now focuses on two questions: whether the slope of the relation changes with redshift and whether the dispersion can be reduced by adding additional parameters to the fit. For both issues there are claims of positive evidence: (i) there is a hint of a drift in the distribution of the light curve stretch parameter as a function of redshift that may bias the SN calibration [107] and (ii) it is found that SNe Ia in younger environments, even after light-curve standardisation, are fainter than those in older ones [108]. The current wisdom is that neglecting these effects is a small contribution to the current level of systematics uncertainties of SN Ia cosmology. However, they will become important for planned experiments aimed to measure the properties of dark energy, in particular with the Vera Rubin Telescope or with the Nancy Grace Roman Space Telescope 6 for which statistics will be excellent.

Extinction
A major uncertainty in the SN calibration is related to the extinction correction that includes two components: the Galactic extinction and due to intervening dust in the host galaxy.
For the Galactic component, the common approach is to use estimates derived from infrared all-sky maps [109] recalibrated on the colour of stars with spectra obtained by the Sloan Digital Sky Survey [110] (c.f. NASA/IPAC Extragalactic Database, NED 7 ).
For the Galactic extinction, we usually assume a reddening law with a total-toselective extinction ratio R V = 3.1. 8 In fact, although R V is known to vary with the line of sight, the most recent measurements suggest that the dispersion is relatively small [111], except towards the inner Milky Way [112].
For the host galaxy extinction, the standard approach is to exploit the observed multi-band photometry of the SNe, assuming some homogeneity in their intrinsic colours. In fact, it was observed early-on that the distribution of the B-V colour at maximum for a sample of SNe I has a blue ridge taken to represent the limit for no extinction [81]. Assuming that SNe I with redder colours are affected by dust extinction, their colour excess can be translated into an extinction correction [113]. Actually, with a much better statistics for a clean sample of SN Ia and with improved photometric accuracy, the assumption of a unique colour at maximum fails, in particular, when faint SNe Ia are included. The dispersion was found to be much lower for the B-V color measured at a late phase (30 ÷ 90 days after maximum light) [114] and this was a contribution to the refinement of the decline rate vs. luminosity relation [91].
There is indeed a correlation between the SN intrinsic colour and the luminosity (first noticed in theoretical models [115]) that appears to mimic the extinction reddening law. From one side this means that, with SN photometry alone, it is difficult to separate the two effects. On the other hand, this gives the possibility to achieve a standardisation of SN Ia luminosity regardless of the physical reason for the color vs. magnitude relation (e.g. the MLCS method [95]).
A debated issue in the literature is the universality of the extinction law. The first claim that R λ for SNe is smaller than the standard value dates back to [116]. Indeed, it was suggested that the dispersion in the SN Ia Hubble diagram is reduced adopting R B ∼ 2 [117], a result that was qualitatively confirmed in a subsequent, extended analysis [118]. With more data, it was found that highly extinguished objects typically show a lower value of R V [119] or, otherwise, a correlation was found between R V and the color excess E(B V ), such that a larger colour excess tends to be associated with a lower value of R V [94]. This was attributed to the effect of multiple scattering as demonstrated via Monte Carlo simulations of the light propagation in circumstellar dust clouds [120].
The extinction law issue remains debated with conflicting claims, some confirming low values of R V = 1.8 − 1.9 [121] while others recovering a dust law consistent with the Milky Way average, R V = 2.9 [122]. A most recent analysis, based on the largest SN Ia sample assembled so far, finds again evidence for variations of R V that, as a further complication, appear to correlate with the galaxy Hubble type. This may explain the apparent diversity of SN Ia luminosity in host galaxies of different Hubble types even after standardisation, which would then be related to differences in the dust properties and not to intrinsic differences of the progenitor systems [123].
To disentangle dust reddening from intrinsic colour differences, it would be useful to have an independent measure of extinction. In this respect, an opportunity comes because interstellar gas and dust are mixed. This explains why measurements of the equivalent width (EW) of interstellar absorption lines of NaI an KI in galactic stars strongly correlates with extinction [124], although with significant dispersion for high reddening. While it is found that, in general, to a high extinction corresponds a large EW of NaID lines even when measured in low-resolution spectra [125], the dispersion of the correlation is much lower when the lines are measured from higher resolution spectroscopy [126]. A detailed analysis found that one quarter of the events display anomalously large EW(NaID) with respect to the amount of dust extinction derived from their colours [127] . This is attributed to an over-abundance of Na in their circumstellar medium. In addition, for some well monitored SN Ia, it was found that NaID EWs vary with time [128]. The interpretation here is that the variable SN radiation field affects the ionisation conditions in the CSM, confined within a short distance from the SN. In turn, the presence of a dense, local CSM was taken to support the single degenerate scenario for SN Ia explosion. Yet, to date, variable NaID lines was found only for a small sample of SNe Ia (eg. [129]). Overall, the measurement of NaID is not providing sufficiently accurate estimates of the extinction, at least in the context of accurate distance determinations.
An alternative approach to reduce the uncertainties is to observe SNe in the NIR, where extinction is much lower. Actually, the first infrared observations of SNe I, while promoting the separation in two different types, Ia and Ib (cf. Sect. 2) also suggested that SNe Ia feature a small dispersion of absolute magnitudes [17]. This result was confirmed by further analyses [130,131] although a (shallow) decline rate vs. luminosity relation was found also in NIR [132]. The latest combined optical/NIR light curves confirm that the ratio of total to selective extinction, as seen for SN Ia is, on average, smaller than the standard value, although with a significant dispersion from object to object [133].
In conclusion, despite the efforts, the extinction correction remains a weakness of the SN luminosity calibration.
To reduce the systematics, the current wisdom is that the qualification of distance indicators is attributed only to "normal" SNe Ia, i.e. those with luminosity decline rate in a limited range (eg. Δm 15 (B) < 1.5 mag) and with observed, blue colour (e.g. 0.2 mag). These requirements guaranty that the intrinsically red fastdecline SN Ia and at the same time, high-extinguished SN Ia are filtered out, reducing the weight of the extinction correction uncertainty.

Calibration of the distance ladder
The absolute calibration of SNe Ia requires a primary distance indicator that can reach galaxies at a distance of 20-30 Mpc, at least. Within this volume, a few tens SNe Ia were observed in the last decades, which allow the required statistics of "local" SNe Ia needed to reduce the random errors.
As I recalled in the introduction, the very first recognition of the enormous luminosity of SNe with respect to ordinary novae was related to the detection of Cepheid variables in a few, very nearby galaxies. However, for many years the instrumentation did not allow to extend the detection of Cepheids to more distant galaxies and reach a fair sample of SN hosts. Then, other indicators were used, such as for instance the average luminosity of the brightest stars in a galaxy [134]). Another approach was to use novae, calibrated in the Andromeda galaxy using RR Lyrae stars, to estimate the distance of the Virgo cluster, where a number of SNe Ia were discovered [117]. Neglecting that the Virgo cluster has an intrinsic distance extension, this approach implies an intrinsic dispersion of about 0.3 mag.
The first derivation of the absolute magnitude vs decline rate relation, for which both good statistics and individual galaxy distances were required, made use of two other calibrators, namely the TFR linking the galaxy luminosity to its rotation velocity [135] and the method based on SBF (eg. [136]). However, the intrinsic dispersion of TFR is relatively large (∼ 0.3 mag [137]) and hence this method was not used in subsequent works for SN Ia calibration. SBF, on the other hand, promises to achieve better precision, with a claimed dispersion of ∼ 0.1 mag [138]. In addition, Cepheids being massive stars, are linked to recent star formation and are not found in earlytype galaxies, while the SBF method is best applied to smooth, early type galaxies. Then, the alternative approaches can be exploited to test possible systematics in SN Ia luminosity calibration, such as its dependence on the properties of the parent stellar population.
An alternative, primary distance indicator that is gaining popularity in the literature is the TRGB measured in the colour-magnitude diagram of the galaxy stellar population [139]. The method can be used in all types of galaxies and it is claimed to provide distances with an uncertainty comparable to Cepheids, ∼ 0.1 mag [140]. Its main advantage is the existence of a universal magnitude limit to the TRGB, based on understood physical ingredients of the stellar evolution theory.
Still, in the current literature, Cepheids maintain a dominant role as primary distance indicators. The main actor in this research has been the Hubble Space Telescope (HST) that was extensively used for the detection and accurate measurement of Cepheid light curves, since its early days of operation. The breakthrough came with the installation of the Wide Field Camera 3 (WFC3), in the last service mission of 2009, that allowed to measure Cepheids in galaxies up to 40-50 Mpc, a distance including a good number of SN Ia hosts. The claim of a Hubble tension (cf. Sect. 9) motivated a renewed effort that, in a few years, has increased the number of SNe Ia calibrated using Cepheids from 19 [74] to 42 [141]. To date, all the known SN Ia hosts in the reach of HST have been observed and, from now on, the statistics can increase only if new SN Ia explodes within the local volume. These nearby SNe Ia however are expected at a rate of only 1 per yr.
To be used as primary distance indicators, cepheids need to be accurately calibrated. Nowadays this is done using three different sources: galactic cepheids, detached eclipsing binaries (DEB) and megamaser in nearby galaxies.
Until recently, accurate parallaxes for galactic Cepheid also required time consuming observations with HST (eg. [142]) and was limited to 15 Cepheids [74]. A major progress was achieved by exploiting the high precision parallaxes of the third GAIA data release that allowed to increase the sample of galactic Cepheids by a factor of five [143].
Detached eclipsing, double-lined spectroscopic binaries offer the opportunity to measure their distance with a method that is essentially geometric, but with the nontrivial need to estimate the surface brightness of each component on the basis of something measurable, like the colour, or the spectral line ratios [144]. With this method, it was possible to measure the distance of the LMC and SMC with a precision of about 1-2% [145,146]. With a similar precision, the distance of NGC 4258 was measured by means of another powerful geometrical method, i.e. the observation and modelling of the Keplerian motion of the masers in the rotating circum-nuclear disk of this Seyfert galaxy [147]. Because of the available accurate distance, this galaxy became target of extensive searches and measurements of Cepheids, now counting to over 400 [141].
After being accurately calibrated, Cepheids have been deeply scrutinised to check for many possible systematics, of which the most important is the extinction correction and the luminosity dependence on the metallicity. In a leading program (SHOES, [74]), the issue of extinction was addressed by the combination of optical and nearinfrared observations . Similarly to what happens with SNe Ia, there is a correlation between colour and luminosity for Cepheids of a given period, with redder Cepheids being fainter. The wisdom is that using a colour correction it is possible to reduce the dispersion of the luminosity regardless of the physical origin of the red color. The dependence of Cepheids luminosity on metallicity was also long debated, with empirical and theoretical estimates often in disagreement. With the contribution of the new GAIA parallaxes and improved LMC and SMC distances, it became possible to perform a crucial empirical test, showing that, in the K band, metal-rich Cepheids are intrinsically brighter that metal poor ones with the same pulsation period [148].
The improved statistics of both geometrically calibrated Cepheids and calibrated SNe Ia, allowed a precise determination of the fiducial SN Ia absolute magnitude, namely the predicted absolute magnitude in the specific standardisation relation. In particular, adopting the following standardisation relation [149]: where x is the stretch parameter, c is a measure of the SN colour, μ the distance modulus, and α = 0.148 and β = 3.112 are parameters derived from the nominal fit of the data, the value of the fiducial SN Ia absolute magnitude, roughly speaking the average absolute magnitude of "normal" SN Ia, is M B = −19.253 ± 0.027 mag [141].
As I will discuss in Sect. 9, the value of the Hubble constant derived with this calibration is in tension with the one deduced from CMB by the Planck mission. In the quest to probe all the possible source of systematics, two techniques mentioned above, TRGB and SBF, are used as alternative methods to extend the Cepheid local scale to the distance of nearby SN Ia host galaxies. The formal consistency between the results obtained with these methods or using Cepheids is still an open issue and requires further investigation.

The Hubble diagram of SN Ia
The history of the Hubble diagram would require a full book. Here I will focus only on the use of SNe and I will arbitrarily pick up what I think are representative results. Additional discussion and references can be found in [150][151][152].
A first plot of the Hubble diagram using SNe I (not yet separated in types Ia and Ib/c) was published in [153] extending to redshift z ∼ 0.03. The plot was used to demonstrate the homogeneity of SNe I (with a measured dispersion of 0.6 mag) and to calibrate their absolute magnitudes on the scale of the still poorly known Hubble constant.
An interesting early attempt was made to derive the SN calibration from physical principles, namely the assumption that the SN I light curves can be modelled with thermal emission from an expanding, optically thick photosphere. This was a modification of the method that Baade introduced for pulsating stars, where the expansion velocity is derived from the spectral lines blueshift and the photospheric temperature from the stellar colour-temperature relation. Later, the same concept was applied with more success to SN IIP (cf. Sect. 7). With SNe Ia, due likely to the coarse approximations, an unusually low value of H 0 = 40 km s −1 Mpc −1 was obtained [154] (to improve readability, hereafter I will skip the units for H 0 ).
In the early '70s, a first comprehensive attempt was made to use SNe I as cosmological probes to test the cosmic expansion [83]. Given the accuracy of the data available at that time, it was really too early to get significant constrain on the expansion history. For the local expansion rate, a value of H 0 = 92 was obtained.
With the conventional approach of the distance scale ladder, especially in the early days, the trade-off was between using many different primary distance indicators, allowing to build a higher statistics, or instead use only one distance indicator, to privilege homogeneity. An example of the latter approach was the use of the brightest red supergiants to measure the distances of just two SN I host galaxies [134] . With the calibrated SNe I, an Hubble diagram including 16 SNe I was produced. Then, a value of H 0 = 50 ± 7 was obtained where, in perspective, the error was certainly underestimated. The lesson that applies also to later studies, is that an apparent homogeneity may hide unknown systematics.
Another approach to build a distance ladder relied on novae to measure the distance of the Virgo cluster, then calibrate the SN I discovered in Virgo and finally use SNe I to estimate the distance of the Coma cluster, H 0 = 70 ± 15 was derived [117]. While the value appears close to the modern estimate, I would rather stress that the precision appears appropriate for the data available at that time.
A Hubble diagram including only SNe Ia, just then recognised as a separate class, was published by [155]. The SN Ia calibration was tied to the Virgo cluster distance, calibrated with many different independent methods, providing a value of H 0 = 46 ± 10. Again the uncertainty was related to that of the primary distance indicators but, also, to the poor accuracy of SN Ia photometry. A major step forward came by exploiting the observational efforts of the Calán/Tololo SN survey that provided accurate CCD photometry for a good sample of SNe Ia in the Hubble flow and a new calibration of the Phillips relation [156]. With the Cepheid calibration of 4 nearby SN Ia, they obtained H 0 = 63.1 ± 3.4 (internal) ± 2.9 (external).
With the new detectors allowing us to reach much fainter targets, it became feasible to detect and measure SNe Ia at redshift z > 0.2 ÷ 0.3. The first successful attempt to detect a distant SN was made with a small telescope and, although requiring a big effort for only one SN Ia at z = 0.31, proved that observational cosmology with SNe was feasible [48]. Then, the focus of SN searches shifted from the measurement of the Hubble constant to probing the cosmic deceleration parameter. In fact, on the basis of the prevalent cosmological scenario of those years, the cosmic expansion rate was expected either to remain constant along the cosmic history or, for a matter density above a critical value, to progressively decrease.
At the beginning of the '90s, two independent groups, first the Supernova Cosmology Project (SCP) [157] and soon after the High-Z team (HZT) [158], set up systematic searches and measurements of high redshift SNe Ia, using the best worldwide observing facilities, that in a few years collected a sample of a dozen SN Ia each.
While the statistics built-up, a basic physical test was successfully performed. By measuring the time dilation of a distant SN Ia, it was proved that the redshift is indeed due to the cosmic expansion motion rather than to other possible explanations, such as photon energy dissipation during travel (tired-light theory) [159].
For what concerns the fate of the expansion, the expectation was that, in case of a significant deceleration, distant SNe Ia had to be brighter than in a Universe with constant expansion. Surprisingly, the observations show that distant SN Ia are fainter than in the uniform expansion prediction. This result being at odd with the expectation, required a deep scrutiny before publication. Indeed, that the two groups found consistent results with a fully independent analysis was influential to convince the broad scientific community [152]. The two papers, published almost simultaneously, were indeed in excellent agreement both providing the first evidence that the cosmic expansion was accelerated [160,161]. Although with SN Ia alone there is a degeneracy in the determination of the cosmological parameters, in particular with the mass density Ω M , it was concluded that the SN Ia data require the contribution to the cosmic dynamics of a cosmological constant Ω Λ or some sort of dark energy.

Other SN types: IIP, SLSN, Kilonovae
Whereas SNe Ia are by far the favoured luminosity distance indicators, the possibility to use other SN types has been also explored. Among core-collapse SNe, the most interesting types are SNe IIP, for which two alternative approaches have been elaborated.
One is an implementation of the expanding photosphere method (EPM) that, as mentioned above, was first applied to SN I. It turns out that EPM is better applied to SN IIP because, in a first approximation, their spectral energy distribution can be fit with a blackbody. This allows to translate the observed luminosity into an angular radius of the photosphere, once an estimate of the temperature is available. The linear radius, on the other hand, can be estimated knowing the time of explosion and the expansion rate of the photosphere, the latter derived from the analysis of the spectral line profiles. The comparison of the angular and linear radii provides the scale factor, i.e. the distance of the SN. Actually, it was soon recognised that the black-body approximation is not good enough and a correction, named "dilution factor", was introduced, at first estimated from empirical considerations [162]. A first attempt to obtain the dilution factor from physical principles, by modelling the radiation transfer in the SN atmosphere, provided an interesting estimate, H 0 = 73 ± 7 [163]. Again, the error was clearly underestimated if we consider that subsequent works, exploiting the same method, produced very diverse values [164]. More recently, efforts have been made to replace the dilution factor correction with detailed spectral modelling (eg. [165]). This new approach promises uncertainties of the order of ∼ 10% with no need for external calibrators, although at the cost of significant observational efforts. In fact, measuring distant SN IIP (z > 0.5), will require the use of the most advanced, new generation telescopes.
An alternative approach to exploit SNe IIP as standardised candles was based on the discovery of a correlation between the ejecta expansion velocities and the plateau luminosity [166]. Here again, the main problem is the absolute calibration through primary distance indicators and the extinction correction. Early estimates based on this approach gave values of H 0 in the range 65-75 with a precision of 15% [167]. Other approaches to SN IIP standardisation were proposed, i.e. considering colors [168]. A most recent analysis, using Cepheids and TRGB to calibrate local SNe II and then a sample of 89 SNe IIP in the Hubble flow, produced a result, H 0 = 75.8 ± 5.0 [169], very similar to that derived from SN Ia although with less precision. Unfortunately, since the redshifts of the current sample of SNe IIP extend only to z = 0.2, with just few events reaching z = 0.4, at present SNe IIP cannot be used to probe the cosmic acceleration. This is why researchers began to explore the possible use of SLSNe (cf. Sect. 2) that, being a hundred time more luminous than SNe Ia, can be seen at a much larger distance, possibly up to z ∼ 4 [170]. For redshift z > 1, SLSNe have also the advantage of a higher UV flux compared to SN Ia. Indeed, the high abundance of heavy elements in the ejecta of SN Ia causes a higher opacity at short wavelength. At present, the focus is on the hydrogen-free SLSNe I, for which standardisation methods seem to give promising results [171]. There are two issues with the use of SLSNe: (i) the actual dispersion of their absolute magnitude after standardization is still debated and (ii) since they are very rare, 1 out of 1000 SNe, they cannot be calibrated in the local Universe using primary distance indicators. Therefore, they are best used to probe the large scale evolution of the cosmological parameters, rather than the local value of the Hubble constant. With this caveat, the first attempt to plot a Hubble diagram of SLSNe-I extending to redshift z = 2 returned an estimate of the dark energy contribution consistent with that found with SNe Ia. [172] For the sake of completeness, I mention also the case of kilonovae (KNe), bright transients associated with the merging of two neutron stars (NS). 9 To date, only one case of KN with complete multi-messenger observations was found, AT2017gfo, discovered after the detection of the gravitational wave transient GW170817 and the associated short GRB [173]. KNe are attractive distance indicators because the analysis of GW signal, adopting general relativity provides, along with the source parameters, their luminosity distance, with the main uncertainty being the binary orbital inclination angle. This method does not need primary distance calibrators and the KN electromagnetic emission is only used to identify the host galaxy and hence, measure the redshift. From the single case of GW170817, it was already possible to derive an estimate of the distance of the host galaxy that, although still with a large error, is in excellent agreement with other estimates, providing a proof of concept, along with a value of H 0 = 70 ± 10 [174].
The excellent precision on the distance estimate that can be achieved by the modelling of GW signals promoted the elaboration of statistical techniques to obtain the Hubble constant even in absence of electromagnetic counterpart detection [175]. This approach will become more and more useful with the increasing number of GW detections at a larger distance (the latter help to reduce the dispersion due to the galaxy peculiar motion). Conversely, it has been claimed that, through the modelling of the electromagnetic emission, it is possible to standardise the KN optical luminosity even without GW data [176]. Considering the improving sensitivity of GW detectors and the new frontiers for optical transient search of the Rubin telescope, the prospects in this field are promising. 9 KNe are not usually aggregated to the general class of SNe, although they share many common features. They are luminous transients that result from a terminal event in a binary stellar system. The merging of two degenerate stars (NS) from which they originate reminds the merging of WDs that produce SN Ia. They are rare, roughly ten times less frequent than SNe Ia. The light curve is explained by the ejection of a gas cloud at least partly powered by radioactive decay of nuclear reaction ashes. That is, in many respects, they could be considered a subclass of SNe although, the neutron stars from which they originate are, themselves, relics of SN explosion.

Dark energy
The quest to confirm the unexpected discovery of dark energy prompted a renewed effort to search for distant SNe, as well as of possible systematic biases. A detailed scrutiny of the possible systematics effect on the luminosity of high redshift SNe Ia, focused on the following contributions: -the possible effect of grey dust, causing absorption but not reddening. Modelling the opacity of the Universe as a function of redshift, it was concluded that the uncertainty propagated to the cosmological parameters is only a few percent [177]. -weak gravitational lensing by intervening matter along the line of sight may cause the SN luminosity magnification or, otherwise, de-magnification depending on the lensing convergence. It is expected that the average luminosity of the SNe Ia does not change significantly, but the dispersion increases. This effect seems barely detected on existing data [178] -an intrinsic evolution with redshift of the SN Ia standardization parameters. While the current analyses do not show evidence of such effect [179], it needs to be verified for all new SN Ia samples.
As I mentioned above, the SN searches became more efficient with the introduction of CCD mosaic detectors that, by increasing the field of view, allowed to reduce the number of telescope pointings needed to cover a certain sky area. In turn, this allowed to increase the exposure time for each pointing and gave access to a much larger volume, bursting the number of detected SN candidates. Examples of the typical observations required for SN discovery, light curve measurements and spectroscopic classifications are shown in Fig. 7. I may stress that, nowadays, the main limitation is the time allocation of large telescopes needed for the spectroscopic classification. The other critical limit comes from the fact that for z > 1 the peak of the SN Ia spectral energy distribution moves to the NIR. Measuring the light curves of these faint NIR transients is hardly feasible for ground-based facilities and requires an intensive use of the Hubble Space Telescope (HST). This explains the relatively small number of events at these redshifts.
The quest to secure a sample of SNe with good statistics and distributed over the widest possible range of redshifts, prompted the compilation of data from different surveys, eg. the Union compilation with 307 SNe Ia [180], the SLSN sample (474 SNe Ia) [181], the JLA compilation [182] (740 SNe Ia), the Pantheon sample [179] (1048 SNe Ia), up to the latest Pantheon+ sample [149] including 1530 selected SNe Ia. Because the data were collected and analysed by different groups, it was important to ensure accurate relative flux calibration, accounting for differences in instrumental setups and performing uniform data analysis, including for the SN light curve fitting and standardisation modelling [182].
All the new analyses confirmed the early claim of an acceleration of the cosmic expansion rate, with a progressively smaller uncertainty.
An example of a modern version of the Hubble diagram for SNe Ia is shown in Fig. 8. The good statistics for SNe Ia are apparent in the Hubble flow up to z ∼ 1. As I mentioned above, for local SNe Ia we are limited by the intrinsic low event rates, whereas at higher redshifts the limitation is the available instrumentation. It appears  [179], https://archive.stsci.edu/prepds/ps1cosmo/) whereas low redshift SNe Ia are retrieved from [74] (Riess16, calibrated with Cepheids), [183] (Khetan21, calibrated with SBF) and [184] (Anand21, calibrated with TRGB). The distance moduli are computed for a flat cosmology with Ω Λ = 0.7 and adopting the [70] calibration of nearby SN Ia (H 0 = 73.2). The bottom panel shows the residuals with respect to the adopted cosmology. The dot-dashed line is the expected trend for a null cosmological constant. Instead the dashed line at redshift z < 0.01 illustrates the shift of the Planck H 0 calibration with respect to local SN Ia calibration that the difference on the SN distance moduli with or without dark energy is maximum at about z ∼ 1, and is about 0.25 mag. This is similar to the typical uncertainty of high redshift SN photometry that is why good statistics is needed. The current data are consistent with the expected smaller deviation at z > 1. It is important to confirm this result with better statistics since it can exclude systematics effects different from a dark energy contribution.
It is well known that when it comes to the estimate of certain combination of cosmological parameters, SNe Ia alone shows a high degeneracy. The degeneracy was broken by the observation of the cosmic microwave background (CMB) anisotropies obtained by the WMAP experiment [185] and by the Planck satellite [186].
The combination of SN Ia and CMB complemented the constraints from the Barionic Acoustic Oscillation (BAO) [187,188], are the empirical basis of the current paradigm of a flat, expanding Universe whose main constituents are cold dark matter and a cosmological constant, referred to as the standard ΛCDM model, with Ω M ∼ 0.3 and Ω Λ ∼ 0.7. The six-parameters fit of the CMB provided also an estimate of the Hubble constant although with significant degeneracy with other parameters, in particular Ω m . The degeneracy can be broken either fixing some of the fit parameters, e.g. setting a flat geometry, or using other cosmic probes, With the latter approach, Plank data provided an estimate of H 0 = 67.3 ± 1.2, that is in tension with the direct estimate from SNe Ia. While the measure of cosmic acceleration does not depend on the actual value of the Hubble constant, the tension, if confirmed, will point to a failure of the standard cosmology model. This will be further discussed in Sect. 9. The debate on the nature of dark energy began with the first evidence for an accelerated expansion. Possible explanations range from the contribution of vacuum energy due to quantum fluctuations, to possible modifications of the standard gravity laws. Each physical explanation produces a specific cosmological model, for which parameters can be derived by the fit to the observables, with the warning that, as for all model fitting, a model with more parameters may give a better fit to the data, though this does not necessarily prove that the model is correct [189].
Current data allow only a few steps in the exploration of the nature of the dark energy, typically based on a parametrisation of the dark energy properties and their possible evolution with time.
In particular, a first generalisation of the ΛCDM comes with the introduction of the parameter w = p/ρ, describing the ratio between the pressure and the energy density of dark energy, the so-called equation of state of dark energy. To produce an accelerated expansion, w must be negative and smaller than −1/3 (the limit for an empty accelerated Universe). A recent example of the confidence contours for the cosmological parameters w and Ω M are shown in Fig. 9 from [182]. The great improvements achieved, thanks both to the increasing of the SN Ia statistics and to the higher resolution and signal to noise of the CMB map, is clearly seen in the comparison of the area enclosed in the dashed contours and in the filled area with the same colour. It is also clear that a strong degeneracy in the cosmological parameters for any given probe taken alone does exist. However, since the different probes have different projections in the w vs. Ω m plane, the combination of SNe Ia and CMB provides a fairly tight constraint (grey area in the figure), fully consistent with a value of w = −1 appropriate for a cosmological constant component. This result was confirmed, with better and better precision in recent literature, eg. w = 1.007 ± 0.089 in [179].
A key to understand the physics of dark energy is to constraint its possible time evolution. A typical parametrisation for a possible evolution is ω(a) = ω 0 +ω a (1+a) with a = 1/(1 + z). Although I have to stress that at present the uncertainties are still fairly large, the available observations are consistent with ω a = 0, suggesting that there is no evolution in the equation of state of dark energy [179].
To address this question and, in general, for a better discrimination of the alternative cosmologies, new experiments have been proposed, each with a focus on some specific probes. Among these, the most ambitious project that involves SN Ia is the Roman Space Telescope, which, by means of a wide field infrared survey, will allow the discovery of thousands of SNe Ia up to redshift z ∼ 2.0 and set constraints to ω 0 of ∼ 1% and, most important for the cosmological model implication, to ω a of ∼ 10% [190]. 10

Hubble tension
In the last few years, the tension of H 0 became the focus of the distance ladder discussions.
The Hubble constant is a fundamental cosmological parameter for which over a thousand estimates have been published in a century [192]. The initial estimates of Lemaître and Hubble (H 0 = 500 − 600) were based on severely underestimated distances and the subsequent H 0 estimates progressively reduced until, by the early '60, the typical values were < 100.
Then, at least for three decades, there were few apparent progresses with most estimates remaining polarised on either a long or a short scale, and still in the early '90s the reported values range between H 0 = 45 ± 3 [193] and H 0 = 90 ± 10 [194]. In retrospective, it is interesting to note that the modern estimates are very close to the average of these early estimates for which, we have to admit, the uncertainties were definitely underestimated.
A decade ago, it appeared that the debate on the long/short Hubble scale was finally going to settle. The value derived using SNe Ia, H 0 = 73.8 ± 2.4 [195] and from CMB (WMAP at the time), H 0 = 71.0 ± 2.5 [196] were found in excellent agreement.
However, as I mentioned before, a few years later the first results from Planck provided a lower value, confirmed in the final data release with only a minor adjustment to H 0 = 67.4 ± 0.5 [188]. In the same period, the value obtained through SNe Ia calibrated with Cepheids was refined to H 0 = 73.2 ± 1.7 [74]. At this point the CMB and the SN Ia estimates of H 0 were different at the 3σ level.
The shift in magnitude that corresponds to the different values of H 0 is about 0.2 mag. This is shown in Fig. 8 where the dashed line, drawn only for redshift z < 0.01, corresponds to the Planck estimate. It is apparent that the dispersion of distance measurements of local SNe is still fairly large, especially if one considers the systematics related to the use of different calibrators. Note that the uncertainties on redshifts, not shown in the figure, contributes to the dispersion for nearby events, due to the unknown galaxy peculiar motions.
The claim of a H 0 tension between the high redshift estimate deduced from the CMB fit and the low redshift value obtained with SNe Ia motivated renewed efforts of many groups. From the side of the CMB analysis, the claim is that, in a standard ΛCDM cosmology, there is no margin to accomodate for a value of H 0 as large as that found with SN Ia. Therefore the attention focused on the SN Ia estimate, in particular, searching for possible systematics in the calibration. The current status of the debate is illustrated in Fig. 10, showing the value of H 0 obtained with different primary distance calibrators and/or different groups. 11 A thorough comparative analysis of the different methods and results can be found, among others, in [141,192,198]. Here I only note that, as also seen in Fig 10, the analysis using Cepheids as primary calibrators of SNe Ia provide consistently "high" value of H 0 . Instead estimates based on TRGB calibration show a larger dispersion among different authors, but typically smaller H 0 values with respect to Cepheids, in some cases formally consistent with the CMB value. The same considerations may apply to SBF calibration although in this case the estimated errors are still relatively large.
The dispersion of SN Ia measurements in Fig. 10 may be taken as an indication that the individual errors are underestimated. This seems to be confirmed by analyses of the dispersion of distance estimates for galaxies with multiple measurements [199]. However, I have to recognise that the latest studies (cf. [143]) have secured all: (i) solid Cepheid calibration with the crucial GAIA contribution, (ii) good statistics for Cepheid calibrated SNe Ia and (iii) complete analysis of statistical and systematic uncertainties. The conclusion is that the present best estimate of the Hubble constant measured in the local Universe is H 0 = 74.0±1.0 significantly different, 5σ deviation, from the Planck estimate. [143].
If we accept this conclusion, we are left with one option only: the adopted cosmology is incomplete or wrong. This may not be surprising considering that the ΛCDM model includes as main components three ingredients that are not yet explained on the basis of fundamental physics, namely inflation, dark matter and dark energy. We may also remind that there are other tensions on current cosmologies, although possibly less compelling, i.e. the discrepancy on the S8 parameter measured with CMB or with the weak lensing. [197].
Therefore, the implementation of additional parameters, the exploration of alternative cosmologies or of new laws of physics, seem to be required. The choice in this respect is very rich and growing, well beyond the scope of this review. A detailed discussion of the possible alternatives to solve the Hubble tension is presented in [197]. They reached the conclusion that at present no clear direction is prevailing and, quote, "early or dynamical dark energy, neutrino interactions, interacting cosmologies, primordial magnetic fields, and modified gravity provide the best options until a better alternative comes along". . A value base of SBF distances alone is also shown (green). The primary distance indicator used for SN calibration is indicated on the right side. The estimate of Riess-2016, which data are, in fact, included and extended in Riess-2021, are shown to allow the perception of the improvement of the uncertainty achieved in the last years. References are as follow: Plank-2020 [188], Riess-2016 [74], Riess-2021 [141], Hamuy-2021 [198], Freedman-2020 [140], Soltis-2021 [200], Anand-2021 [184], Khetan-2021 [183], Blackslee-2021 [201] 10 Conclusions A century since the identification of SNe, their study remains a hot topic in astrophysics, with new unexpected findings, opening new questions on stellar evolution, explosion mechanisms, remnants properties and multi-messenger links. In this review, I focussed on the use of SNe as cosmological probes and in particular on SN Ia as probes of the cosmic expansion. In this field, two surprising results were found: first, that the cosmic expansion is accelerated and, second, that the current expansion rate parameter measured at low redshift is higher than that at high redshift, as deduced by the cosmology model fit of the CMB.
Based on the most recent literature, it is a consolidate result that the Universe expansion is accelerated due to a still unidentified dark energy. Future experiments that include the use of SNe Ia and, possibly, other SN types will contribute to unveil the nature of dark energy.
On the other hand, for the H 0 tension, the dispersion of measurements in the literature, in some cases obtained with different SN Ia calibrations, may be taken as an indication that the claimed 1% uncertainty is underestimated, which would not be new in the history of H 0 measurements. In this case, the tension would not have a high significance. However, the most recent works including improved Cepheid calibration, good statistics of local SN Ia, and deep scrutiny of all sources of uncertainty, do not show obvious weaknesses.
Because of the consequence of the Hubble tension for the cosmological model, a verification with independent probes is needed. In this context, the GW sources in the new multimessenger era, seems to offer promising opportunities.