These dark circles can likely be attributed to vascular network prominence caused by hollowing of the contents of the orbital rim, pigmentary changes in the periorbital area caused by extravasated hemoglobin and its breakdown products (bilirubin and biliverdin), or to visibly accumulating fluid in the lower eyelid due to local processes such as atopy or systemic fluid retention.
Dark circles under the eyes are very literally representative of infraorbital (or periorbital/periocular) hyperpigmentation. I was able to find literature pretty quickly on this subject dating all the way back to the '60s:
"Dark Circles Under the Eyes in Children." Meyer B. Marks. Clinical Pediatrics. 1966.
Dark
discolorations
or
shadows
under
the
eyes
as
seen
in
some
children
are
often
ascribed
to
fatigue
or
sleeplessness.
With
older
children
and
even
in
adults
a
popular
notion
is
to
attribute
these
to
chronic
eyestrain.
With
an
adolescent
girl
at
menarche
the
mother
may
associate
them
with
menstruation.
Many
of
the
bluish-black
discolorations
in
the
lower
orbitopalpebral
grooves
of
children
– termed "allergic
shiners" – result
from
long-
standing
perennial
nasal
allergy.
They
are
rarely
seen
in
the
first
year
of
life
unless
the
infant
had
nasal
obstruction
with
accompanying
mouth-breathing
from
birth.
Children
with
uncomplicated
seasonal
allergic
rhinitis
do
not
have
allergic
shiners.
This early paper (which mentions allergies as the cause of these dark circles) alludes to sleeplessness or fatigue as being another cause of infraorbital hyperpigmentation. Skipping several decades of research and progress, there are a number of useful, modern review papers on this subject (emphasis mine):
"What causes dark circles under the eyes?" Fernanda Magagnin Freitag,
Tania Ferreira Cestari. Journal of Cosmetic Dermatology. 2007.
Dark rings under the eyes are defined as bilateral, round,
homogeneous pigment macules on the infraorbital regions.
There is no doubt that they are worsened by general
fatigue, especially lack of sleep. This idea is corroborated
by the daily fluctuation of the lesions intensity, according
to the patient status. For this reason, they have been
regarded as a mere physiologic phenomenon.
Dark circles are more pronounced in certain ethnic groups and
are also frequently seen in multiple members of the same
family. These hereditary observations raise a question:
are there any anatomic or histological characteristics
in these populations that could give us a reasonable
etiologic explanation?
Histological characteristics of infraorbital darkening
suggest that they are caused by multiple etiologic factors
that include dermal melanin deposition, postinflammatory
hyperpigmentation secondary to atopic or allergic contact
dermatitis, periorbital edema, superficial location of
vasculature, and shadowing due to skin laxity.
Despite its prevalence and cosmetic importance, there are
few published studies in the scientific literature about
dark circles. Even a good definition of this condition is
lacking. We think the term
infraorbital ring-shaped melanosis
proposed by Watanabe et al. does not encompass its
etiology in a global manner.
As there is neither a general understanding about
dark circles pathogenesis nor a consensus about the
major responsible features, treatments are chosen in a
simplified way, rendering suboptimal results most of
the time. It is important to identify the specific anatomic
problem of each patient in order to individualize treatment.
This gets closer to a direct answer and offers several possible etiologies (bolded) for infraorbital hyperpigmentation. These are further clarified in the context of sleep by publications from this year:
"A study of epidemiological, etiological, and clinicopathological factors in periocular hyperpigmentation." M Chatterjee, B Suwal, A Malik, B V. Journal of Cosmetic Dermatology. 2018.
Conclusion:
Periocular darkening was predominantly not due to pigment, but rather due to cutaneous laxity and vascular visibility through
thin skin. Most of them with pigment had it in the dermis. Clinical dermal pigmentation correlated well with histology, unlike epidermal
pigmentation. Iron and amyloid were not significant as etiological factors in our patients.
"Periorbital hyperpigmentation − An overview of the enigmatous condition."
Manju Daroach, Muthu S Kumaran. Pigment International. 2018.
Periocular dark circles may appear because of various
anatomic factors such as architecture of facial ligaments,
the bony facial structure, midface soft tissue including the
prominence of the orbicularis oculi muscle. Because of ageing,
there is a loss of facial fat leading to inflexible ligaments
causing hollowing effect on orbital rim. There is worsening of
shadowing duetohollowness,which is mainly seen in the tear
trough area in inferomedial orbit. The thin eyelid skin
contributes to the prominence of the underlying soft tissue
and subcutaneous vascular network and the orbicularis oculi
muscle, due to which the overlying skin appears dark.
Visible pigmentary changes in periorbital area may be due to
extravasated hemoglobin and its breakdown products
bilirubin and biliverdin. A variety of pathologic and
age-related processes result in the increased permeability
of the local vasculature resulting in these pigmentary
changes. The lower eyelid tissues may have an increased
tendency to accumulate fluid due to local processes such as
atopy as well as systemic fluid retention and can be limited
inferiorly by the orbital rim because of the cutaneous
ligaments. This fluid often takes on a purplish color due
to the prominent role of the orbicularis muscle in the lower
eyelid. Medical disorders including disorders of liver,
heart, thyroid or kidney, hereditary blood disorders,
vitamin K deficiency.
For a more complete analysis of the different contributing factors to infraorbital dark circles, check out this review from 2016:
"Infraorbital Dark Circles: A Review of the Pathogenesis, Evaluation and Treatment." Ivan Vrcek, Omar Ozgur, and Tanuj Nakra. Journal of Cutaneous and Aesthetic Surgery. 2016.
Infraorbital dark circles are caused by a variety of anatomic features, with contributions from the skin, subcutaneous tissues, orbicularis muscle, vasculature and ligamentous architecture of the lower eyelid and cheek. A detailed understanding of the regional anatomy is crucial to the management of dark circles, which encompasses a wide variety of modalities.
This might not be a totally satisfying answer, but it's not as simple as a single source of pigmentation resulting in dark circles under the eyes. Even though there are many contributing factors, we still use general periorbital hyperpigmentation as an identifier of fatigue, since the various factors (usually) cumulatively represent sleeplessness.